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on a farm. No. tomatoes No. plots ( )f 20–29 329 30– 49 413 50–79 704 80–100 258 a Calculate an estimate of the mean number of tomatoes produced by these plots. b The tomatoes are weighed accurately and their mean mass is found to be 156.50 grams. At market they are sold for $3.20 per kilogram and the total revenue is $50350. Find the actual mean number of tomatoes produced per plot. c Why could your answer to part b be inaccurate? 14 Twenty boys and girls were each asked how many aunts and uncles they have. The entry 4 / 5 in the following table, for example, shows that 4 boys and 5 girls each have 3 aunts and 2 uncles. a Find the mean number of uncles that the boys have. b For the boys and girls together, calculate the mean number of: i aunts ii aunts and uncles/ 0 0 / 0 0 / 0 1/ 0 Aunts 1 0 / 2 3 / 4 1/ 1 0 / 0 7 / 11 Suggest an alternative way of presenting the data so that the calculations in parts a and b would be simpler to make. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 PS 15 A calculated estimate of the mean capacity of 120 refrigerators stored at a warehouse is 348 litres. The capacities are given in the following table. Capacity (litres) No. refrigerators ( )f 160– 12 200– 28 320– 48 400– p 32 A delivery of n new refrigerators, all with capacities between 200 and 320 litres arrives at the warehouse. This causes the mean capacity to decrease by 8 litres. Find the value of n and state what assumptions you are making in your calculations. PS 16 A carpet fitter is employed to fit carpet in each of the 72 guest bedrooms at a new hotel. The following table shows how many rooms were completed during the first 10 days of work. No. rooms completed No. days ( )f 5 2 6 or 7 8 Based on these figures, estimate how many more days it will take to finish the job. What assumptions are you making in your calculations? PS 17 In the figure opposite, a square of side |
8cm is joined edge-to-edge to a semicircle, with centre O. P is 2 cm from O on the figure’s axis of symmetry. X 8 Y Points X and Y are fixed but the position of Z is variable on the shape’s perimeter. 8 P O 36 a Find the mean distance from P to X Y, and Z when angle POZ is equal to: i 180 ° ii 135. ° 6 Z b Find obtuse angle POZ, so that the mean distance from P to X Y, and Z is identical to the mean distance from P to X and.Y EXPLORE 2.3 Six cards, numbered 1, 2, 3, 4, 5 and 6, are placed in a bag, as shown. 15 different pairs of cards can be selected without replacement from the bag. Three of these pairs are {2,3}, {6,4} and {5, 1}. Make a list of all 15 unordered pairs and find the mean of each. We will denote these mean values by X2. Choose a suitable method to represent the values of X2 and their frequencies. Find X2, the mean of the values of X2. Repeat the process described above for each of the following: 1 2 5 4 3 6 ● the six possible selections of five cards, denoting their means by X5 ● the 15 possible selections of four cards, denoting their means by X 4 ● the 20 possible selections of three cards, denoting their means by X3 ● the six possible selections of one card, denoting their means by X1. FAST FORWARD We will study the number of ways of selecting objects in Chapter 5, Section 5.3. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency What does the single value of X6 represent? Do the values of and X5 have anything in common? Can you suggest reasons for any of the common features that you observe? Investigate the values of X r when there are a different number of consecutively numbered cards in the bag. Coded data To code a set of data, we can transform all of its values by addition of a positive or negative constant. The result of doing this produces a set of coded data |
. One reason for coding is to make the numbers easier to handle when performing manual calculations. Also, it is sometimes easier to work with coded data than with the original data (by arranging the mean to be a convenient number, such as zero, for example). To find the mean of 101, 103, 104, 109 and 113, for example, we can use the values 1, 3, 4, 9 and 13. Our x values are 101, 103, 104, 109 and 113, so 1, 3, 4, 9 and 13 are corresponding values of x( – 100). Mean of the coded values is 100) x( Σ − 5 = 1 3 4 9 13 + + + + 5 =. 6 We subtracted 100 from each x value, so we simply add 100 to the mean of the coded values to find the mean of x. FAST FORWARD We will study the standard normal variable, which has a mean of 0, in Chapter 8, Section 8.2. FAST FORWARD ) ( x b We will see how to use coded totals, such as )2 Σ − and x b to find measures of variation in Chapter 3, Section 3.3. ( Σ −, 37 Mean( x ) mean( – 100) 100 106 or x x + = = 100) = x( Σ − 5 + 100 106 = Refer to the following diagram. If we add b– to the set of x values, they are all translated by b– and so is their mean. TIP ( ), b. The If we remove the bracket from Σ −x b we obtain Σ − Σx term Σb means ‘the sum of all the bs’ and there are n of them, so ( Σ − = Σ −. x nb x b ) So, mean translated by –b Values of x − b mean of x − b Values of x mean of x KEY POINT 2.3 For ungrouped data For grouped data These formulae can be summarised by writing = x mean( – ) x b +. b Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 For two datasets coded as (x − a |
) and (y − b), we can use the totals Σx and Σy to find the mean of the combined set of values of x and y. WORKED EXAMPLE 2.9 The exact age of an individual boy is denoted by b, and the exact age of an individual girl is denoted by g. Exactly 5 years ago, the sum of the ages of 10 boys was 127.0 years, so b Σ( − 5) = 127.0. In exactly 5 years’ time, the sum of the ages of 15 girls will be 351.0 years, so g Σ( + 5) = 351.0. Find the mean age today of a the 10 boys b the 15 girls c the 10 boys and 15 girls combined. Answers a = b 127 10 + = 5 17.7 years ( b Σ − 5) b = Σ − b Σ = 127 50 177 and = + × (10 5) 127, so = 177 10 = b We update the boys’ past mean age by addition. Alternatively, we expand the brackets. = 17.7 years. g 351 b = 15 Σ + ( g − = 5 18.4 years We backdate the girls’ future mean age by subtraction. 5) g = Σ + (15 5) × = 351, so Alternatively, we expand the brackets. 38 g Σ = 351 75 − = 276 and g = 276 15 = 18.4 years. c g b Σ + Σ 10 15 + = 177 276 + 25 = 18.12 years WORKED EXAMPLE 2.10 Forty values of x are coded in the following table. 3x – Frequency 0– 9 18– 13 24–32 18 Calculate an estimate of the mean value of x. Answer 39 9) × x = = = 24.45 + (21 13) × 40 + (28 18) × + 3 We calculate an estimate for the mean of the coded data using class mid-values of 9, 21 and 28, and then add 3 to obtain our estimate for x. TIP It is not necessary to decode the values of x – 3. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central |
tendency EXERCISE 2C 1 For 10 values denoted x, it is given that x = 7.4. Find: a Σx b x( Σ + 2) c x( Σ − 1) 2 Twenty-five values of z are such that z( Σ − 7) = 275. Find z. 3 Given q = 22 and ( qΣ − 4) = 3672, find the number of values of q. 4 The lengths of 2500 bolts, x mm, are summarised by ( xΣ − 40) = 875. Find the mean length of the bolts. P 5 Six data values are coded by subtracting 13 from each. Five of the coded values are 9.3, 5.4, 3.9, 7.6 and 2.2, and the mean of the six data values is 17.6. Find the sixth coded value. 6 The SD card slots on digital cameras are designed to accommodate a card of up to 24 mm in width. Due to low sales figures, a manufacturer suspects that the machine used to cut the cards needs to be recalibrated. The widths, w mm, of 400 of these cards were measured and are coded in the following table, where x w= – 24. w – 24 (mm) –0.15 << x –0.1 –0.1 << x 0 0 << x 0.1 0.1 << x 0.2 No. cards ( )f 32 360 6 2 a Suggest a reason why the widths have been coded in this way. b What percentage of the SD cards are too wide to fit into the slots? c Use the coded data to estimate the mean width of these 400 cards. 7 Sixteen bank accounts have been accidentally under-credited by the following amounts, denoted by x$. 917.95 917.98 918.03 917.97 918.01 917.94 918.05 918.07 918.02 917.93 918.01 917.88 918.10 917.85 918.11 917.94 To calculate x manually, Fidel and Ramon code these figures using x( – 917) and x( – 920), respectively. Who has the simpler maths to do? Explain your answer. PS 8 Throughout her career, an athlete has been timed in 120 of her 400-metre races. Her times, denoted by t seconds, were |
recorded on indoor tracks 45 times and are summarised by Calculate her average 400-metre running time and comment on the accuracy of your answer., and on outdoor tracks where 60) 83.7 =. 38.7 t( Σ − t( Σ − 65) = − P 9 All the interior angles of n triangular metal plates, denoted by y°, are measured. a State the number of angles measured and write down the value of y. b Hence, or otherwise, find the value of y( Σ − 30). 39 FAST FORWARD You will study the mean of linear combinations of random variables in the Probability & Statistics 2 Coursebook, Chapter 3. 10 A dataset of 20 values is denoted by x where Σ − values is denoted by y where Σ − y( 2) = 1) 58. Another dataset of 30 36. Find the mean of the 50 values of x and y. x( = Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 11 Students investigated the prices in dollars ($) of 1 litre bottles of a certain drink at 24 shops in a town and at 16 shops in surrounding villages. Denoting the town prices by t and the village prices by v, the students’ data are summarised by the 0.56. totals Σ − 1.1) 1.44 and ( v Σ − 1.2) = = ( t Find the mean price of 1 litre of this drink at all the shops at which the students collected their data. A set of data can be coded by multiplication as well as by addition of a constant. Suppose the monthly take-home salaries of four teachers are $3600, $4200, $3700 and $4500, which have mean x = $4000. What happens to the mean if all the teachers receive a 10% increase but must pay an extra $50 in tax each month? To find their new take-home salaries, we multiply the current salaries by 1.1 and then subtract 50. KEY POINT 2.4 For ungrouped data, x = 1 a Σ ( ) ax b − n + b. |
For grouped data, x = 1 a Σ ( ) ax b f − f Σ + b. The new take-home salaries are $3910, $4570, $4020 and $4900. + + 3910 4570 4020 4900 + 4 = $4350. The mean is The original data, x, has been coded by multiplication and by addition as x1.1 – 50. The mean of the coded data is 4350, which is equal to (1.1 4000) – 50, where × 4000. x= Data coded as ax b– has a mean of ax b−. To find x from a total such as ax b ‘ b– ’ and undo ‘ a× ’, in that order. That is: Σ ( −, we can find the mean of the coded data, then undo ) 40 x = (4350 50) 1.1 or + ÷ 1 1.1 × (4350 50) + = 4000. These formulae can be summarised by writing 1 a [mean( × = x ) ax b − + ]. b TIP ) ax b − can be Σ(. rewritten as a x nb Σ − WORKED EXAMPLE 2.11 The total area of cloth produced at a textile factory is denoted by xΣ and is measured in square metres. Find an expression in x for the area of cloth produced in square centimetres. Answer 1m 100 cm = 2 1m 100 cm 10 000 cm = = 2 2 2 Total area, in square centimetres, is Σ 10 000 x or 10 000Σ. x WORKED EXAMPLE 2.12 For the 20 values of x summarised by (2 xΣ − 3) 104, find x. = We convert the measurements of x from m2 to cm2. Answer 104 20 x = = 5.2 5.2 3 + 2 = 4.1 We first find the mean of the coded values. Knowing that x − = then undo the ‘ 2× ’, in that order, to find x. 3 5.2, we undo the ‘ – 3’ and 2 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - |
Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency Alternatively, we can expand the brackets in − us to find the value of xΣ. 3), which allows (2 xΣ FAST FORWARD We will see how to use coded totals such as ( Σ measures of variation in Chapter 3, Section 3.3. ( ax b Σ − )2 ax b − and to find ) Σ (2 x − 3) 104 = 2 x Σ − (20 3) 104 = × x Σ = 82 x = 82 20 = 4.1 EXERCISE 2D 1 The masses, x kg, of 12 objects are such that x = 0.475. Find the value of Σ 1000 x and state what it represents. 2 The total mass of gold extracted from a mine is denoted by xΣ, which is measured in grams. Find an expression in x for the total mass in: a carats, given that 1 carat is equivalent to 200 milligrams b kilograms. 3 The area of land used for growing wheat in a region is denoted by wΣ hectares. Find an expression in w for the total area in square kilometres, given that 1 hectare is equivalent to 10 000 m2. 4 Speeds, measured in metres per second, are denoted by x. Find the constant k such that kx denotes the speeds in kilometres per hour. 5 The wind speeds, x miles per hour (mph), were measured at a coastal location at midday on 40 consecutive days and are presented in the following table. 41 Speed ( mph) x No. days ( )f 15 < x 17< 17 < x 20< 20 < x 24< 24 < x 25< 9 13 14 4 Abel wishes to calculate an estimate of the mean wind speed in kilometres per hour (km/h). He knows that a distance of 5 miles is approximately equal to 8km. a Explain how Abel can calculate his estimate without converting the given boundary values from miles per hour to kilometres per hour. b Use the wind speeds in mph to estimate the mean wind speed in km/h. 6 Given that 15 values of x are such that (3 xΣ − 2) = 528, find x and find the value of b such that Σ (0.5 x b − ) 138. = 7 For 20 values of y, it |
is given that Σ ( ax b − ) = 400 and Σ ( bx a − ) = 545. Given also that x = 6.25, find the value of a and of b. P 8 The midpoint of the line segment between A and B is at (5.2,–1.2). Find the coordinates of the midpoint after the following transformations have been applied to A and to B. a T: Translation by the vector 7 − 4. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 b E: Enlargement through the origin with scale factor 5. c Transformations T and E are carried out one after the other. Investigate whether the location of the mid- point of AB is independent of the order in which the transformations are carried out. M PS 9 Five investors are repaid, each with their initial investment increased by %p plus a fixed ‘thank you’ bonus of $q. The woman who invested $20 000 is repaid double her investment and the man who invested $7500 is repaid triple his investment. Find the total amount that the five people invested, given that the mean amount repaid to them was $33000. Do you think the method of repayment is fair? Give a reason for your answer. 10 One of the units used to measure pressure is pounds per square inch (psi). The mean pressure in the four tyres of a particular vehicle is denoted by x psi. Given that 1 pound is approximately equal to 0.4536kg and that 1 metre is approximately equal to 39.37 inches, express the sum of the pressures in the four tyres of this vehicle in grams per cm2. 2.3 The median You will recall that the median splits a set of data into two parts with an equal number of values in each part: a bottom half and a top half. In a set of n ordered values, the median is at the value half-way between the 1st and the nth. Consider a DIY store that opens for 12 hours on Monday and for 15 hours on Saturday. The numbers of customers served during each hour on Monday and |
on Saturday last week are shown in the following back-to-back stem-and-leaf diagram. 42 Monday (12) Saturday (15 Key: 0 2 2 represents 20 customers on Monday and 22 customers on Saturday To find the median number of customers served on each of these days, we need to find their positions in the ordered rows of the back-to-back stem-and-leaf diagram. from left to right. The median is at the For Saturday, there are n = 15 values arranged in ascending order from top to bottom and n 1 + 2 In the first row, we have the 1st to 4th values, and in the second row we have the 5th to 10th values, so the 8th value is 38. 15 1 + 2 value. 8th th ( ) = = The median number of customers on Saturday was 38. For Monday, there are n = 12 values arranged in ascending order from top to bottom and from right to left. The median is at the value, so we locate the th 6.5th n 1 + 2 12 1 + 2 th ) ( = = median mid-way between the 6th and 7th values. In the first row, we have the 1st to 6th values and the 6th is 28. The first value in the second row is the 7th value, which is 30. The median number of customers on Monday was 28 30 + 2 = 29. KEY POINT 2.5 For n ordered values, the median is at the n 1 + th value. 2 For even values of n, the median is the mean of the two middle values. TIP We can find the 8 th value by counting down and left to right from 22 or by counting up and right to left from 49. TIP Take care when locating values at the left side of a back- to-back stem-and-leaf diagram; they ascend from right to left, and descend from left to right, as we move along each row. When data appear in an ordered frequency table of individual values, we can use cumulative frequencies to investigate the positions of the values, knowing that the median TIP is at the n 1 + 2 th value. n is equal to the total frequency f |
Σ. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency WORKED EXAMPLE 2.13 The following table shows 65 ungrouped readings of x. Cumulative frequencies and the positions of the readings are also shown. Find the median value of x. x 40 41 42 43 44 f 11 23 19 8 4 cf 11 34 53 61 65 Positions 1st to 11th 12th to 34th 35th to 53rd 54th to 61st 62nd to 65th Answer Median value of x is 41. The total frequency is 65, and n + 2 65 1 + 2 = = 1 33, so the median is at the 33rd value. From the table, we see that the 12th to 34th values are all equal to 41. Estimating the median In large datasets and in sets of continuous data, values are grouped and the actual values cannot be seen. This means that we cannot find the exact value of the median but we can estimate it. The method we use to estimate the median for this type of data is by reading its value from a cumulative frequency graph. We estimate the median to be the value whose cumulative frequency is equal to half of the total frequency. Consider the masses of 300 museum artefacts, which are represented in the following cumulative frequency graph ( 300 250 200 150 100 50 0 1 2 3 4 median 5 Mass (kg) The set of data has a total of n = 300 values. An estimate for the median is the mass of the n 2 = 300 2 = 150th artefact. Copyright Material - Review Only - Not for Redistribution TIP Frequencies must be taken in account here. Although 41 is not the middle of the five values of x, it is the middle of the 65 readings. REWIND We studied cumulative frequency graphs in Chapter 1, Section 1.4. KEY POINT 2.6 43 On a cumulative frequency graph with total frequency n the median is at the n 2 th value. f= Σ, TIP The graph is only an n 2 estimate, so we use rather than to n 1 + 2 estimate the median. This ensures that we arrive at the same position for the median whether we count up from the bottom or down from the top of the cumulative frequency axis. |
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 We draw a horizontal line from a cumulative frequency value of 150 to the graph. Then, at the point of intersection, we draw a perpendicular, vertical line down to the axis showing the masses. Reading from the graph, we see that the median mass is approximately 2.6kg. DID YOU KNOW? The concept of representing many different measurements with one representative value is quite a recent invention. There are no historical examples of the mean, median or mode being used before the 17th century. In trying to find the longitude of Ghanza in modern-day Afghanistan, and in studying the characteristics of metals, the 11th century Persian Al-Biruni is one of the earliest known users of a method for finding a representative measure. He used the number in the middle of the smallest and largest values (what we would call the mid-range) ignoring all but the minimum and maximum values. The mid-range was used by Isaac Newton and also by explorers in the 17th and 18th centuries to estimate their geographic positions. It is likely that measuring magnetic declination (i.e. the variation in the angle of magnetic north from true north) played a large part in the growth of the mean’s popularity. 44 Choosing an appropriate average Selecting the most appropriate average to represent the values in a set of data is a matter for discussion in most situations. Just as it may be possible to choose an average that represents the data well, so it is often possible to choose an average that badly misrepresents the data. The purpose and motives behind choosing an average value must also be considered as part of the equation. Consider a student whose marks out of 20 in 10 tests are: 3,4,6,7,8,11,12,13,17 and 17. The three averages for this set of data are: mode 17, = mean 9.8 and = median 9.5. = If the student wishes to impress their friends (or parents), they are most likely to use the mode as the average because it is the highest of the three. Using either the mean or median would suggest that, on average, the student scored fewer than half marks on these tests. Some of the features of the measures of central |
tendency are given in the following table. TIP Do not confuse the median’s position (150th) with its value (2.6 kg). FAST FORWARD We will use cumulative frequency graphs to estimate the quartiles, the interquartile range and percentiles in Chapter 3, Section 3.2. FAST FORWARD The mid-range, as you will discover in Chapter 3, Section 3.2, is not the same as the median. Advantages Unlikely to be affected by extreme values. Useful to manufacturers that need to know the most popular styles and sizes. Can be used for all sets of qualitative data. Disadvantages Ignores most values. Rarely used in further calculations. Takes all values into account. Frequently used in further calculations. The most commonly understood average. Can be used to find the sum of the data values. Cannot be found unless all values are known. Likely to be affected by extreme values. Mode Mean Median Can be found without knowing all of the values. Relatively unaffected by extreme values. Only takes account of the order of the values and so ignores most of them. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency As an example of the effect of an extreme value, consider the dataset 40,40,70,100,130 and 250. If we increase the largest value from 250 to 880, the mode and median are unchanged (i.e. 40 and 85), but the mean increases by 100% from 105 to 210. Although the median is usually unaffected by extreme values, this is not always the case, as the Libor scandal shows. DID YOU KNOW? LIBOR (London Interbank Offered Rates) are average interest rates that the world’s leading banks charge each other for short-term loans. They determine the prices that people and businesses around the world pay for loans or receive for their savings. They underpin over US $450 trillion worth of investments and are used to assess the health of the world’s financial system. A B C D 2.6% 2.8% 3.0% 3.1% LIBOR = 2.9% The highest and lowest 25% of the daily rates submitted by a small group of leading banks |
are discarded and a LIBOR is then fixed as the mean of the middle 50%. The above diagram shows a simple example. Consider how the LIBOR would be affected if bank D submitted a rate of 2.5% instead of 3.1%. Several leading banks have been found guilty of manipulating the LIBOR by submitting false rates, which has so far resulted in them being fined over US $9 billion. You can find out more about the LIBOR scandal by searching news websites. Consider the number of days taken by a courier company to deliver 100 packages, as given in the following table and represented in the bar chart. No. days No. packages ( )f 1 10 2 40 3 25 4 14 5 8 6 3 45 40 30 20 10 No. days 5 6 A curve has been drawn over the bars to show the shape of the data. The mode is 2 days. The median is between the 50th and 51st values, which is 2.5 days. The mean = (1 10) × + (2 40) × + (3 25) × + 100 (4 14) × + (5 8) × + (6 3) × = 2.79 days Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 The mean is the largest average and is to the side of the curve’s longer tail. The mode is the smallest average and is to the side of the curve’s shorter tail. The median is between the mode and the mean. A set of data that is not symmetrical is said to be skewed. When the curve’s longer tail is to the side of the larger values, as in the previous bar chart, the data are said to be positively skewed. When the longer tail is to the side of the smaller values, the data are said to be negatively skewed. Generally, we find that: Mode < median < mean when the data are positively skewed. Mean < median < mode when the data are negatively skewed. EXERCISE 2E FAST FORWARD In Chapter 3, we will use a measure of central tendency and a measure of variation to better describe the values in a set of data. In Chapter 8, we will study sets of data called normal distributions |
in which the mode, mean and median are equal. 1 The number of patients treated each day by a dentist during a 20-day period is shown in the following stem-and-leaf diagram Key: 1 5 represents 15 patients a Find the median number of patients. 46 b On eight of these 20 days, the dentist arrived late to collect their son from school. If they decide to use their average number of patients as a reason for arriving late, would they use the median or the mean? Explain your answer. c Describe a situation in which it would be to the dentist’s advantage to use a mode as the average. 2 a Find the median for the values of t given in the following table. t f 7 4 8 7 9 9 10 14 11 16 12 41 13 9 b What feature of the data suggests that t is less than the median? Confirm whether or not this is the case. 3 a Find the median and the mode for the values of x given in the following table. x f 4 14 5 13 6 4 7 12 8 15 b Give one positive and one negative aspect of using each of the median and the mode as the average value for x. c Some values in the table have been incorrectly recorded as 8 instead of 4. Find the number of incorrectly recorded values, given that the true median of x is 5.5. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency 4 The following graph illustrates the times taken by 112 people to complete a puzzle. 120 ( 80 40 0 2 4 6 8 10 Time (min) Estimate the median time taken. The median is used to divide these people into two groups. Find the median time taken by each of the groups. a b 5 The masses, m kilograms, of 148 objects are summarised in the following table. Mass ( kg)m m 0< m 0.2< m 0.3< m 0.5< m 0.7< m 0.8< cf 0 16 28 120 144 148 47 Construct a cumulative frequency polygon on graph paper, and use it to estimate the number of objects with masses that are: a within 0.1kg of the median b more than 200 g from the median. 6 A teacher recorded the quiz marks of |
eight students as 11, 13, 15, 15, 17, 18, 19 and 20. They later realised that there was a typing error, so they changed the mark of 11 to 1. Investigate what effect this change has on the mode, mean and median of the students’ marks. 7 The following table shows the lifetimes, to the nearest 10 days, of a certain brand of light bulb. Lifetime (days) 90–100 110–120 130–140 150–160 170–190 200–220 230–260 No. light bulbs ( )f 12 28 54 63 41 16 6 a Use upper class boundaries to represent the data in a cumulative frequency graph and estimate the median lifetime of the light bulbs. b How might the manufacturer choose a value to use as the average lifetime of the light bulbs in a publicity campaign? Based on the figures in the table, investigate whether it would be to the manufacturer’s advantage to use the median or the mean. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 8 It is claimed on the packaging of a brand of battery that they can run a standard kitchen clock continuously for ‘at least 150 days on average’. Tests are carried out to find the length of time, t hours, that a standard kitchen clock runs using one of these batteries. The results are shown in the following table. Time (t hours) 3000 < t < 3096 3096 < t < 3576 3576 < t < 3768 3768 < t < 3840 No. batteries ( )f 34 66 117 33 What could the words on the packaging mean? Test the claim by finding the mean, the median and the modal class. What conclusions, if any, can you make about the claim? 9 Homes in a certain neighbourhood have recently sold for $220 000, $242 000, $236 000 and $3500 000. A potential buyer wants to know the average selling price in the neighbourhood. Which of the mean, median or mode would be more helpful? Explain your answer. 10 A study was carried out on 60 electronic items to find the currents, x amperes, that could be safely passed through them at a fixed voltage before they overheat. The results are |
given in the two tables below. Current (x amperes) No. items that do not overheat Current (x amperes) No. items that overheat 0.5 60 0.5 0 a Find the value of p, of q and of r. 1.5 48 1.5 p 2.0 20 2.0 q 3.5 6 3.5 r 5.0 0 5.0 60 b Cumulative frequency graphs are drawn to illustrate the data in both tables. 48 i Describe the transformation that maps one graph onto the other. ii Explain the significance of the point where the two graphs intersect. PS P 11 The lengths of extra-time, t minutes, played in the first and second halves of 100 football matches are summarised in the following table. Extra-time (t min) First halves ( )cf Second halves ( )cf <t 1 24 6 <t 2 62 17 <t 4 80 35 <t 5 92 82 <t 7 97 93 <t 9 100 100 a Explain how you know that the median extra-time played in the second halves is greater than in the first halves. b The first-half median is exactly 100 seconds. i Find the upper boundary value of k, given that the second-half median is k times longer than the first- half median. ii Explain why the mean must be greater than the median for the extra-time played in the first halves. PS 12 Eighty candidates took an examination in Astronomy, for which no candidate scored more than 80%. The examiners suggest that five grades, A, B, C, D and E, should be awarded to these candidates, using upper grade boundaries 64, 50, 36 and 26 for grades B, C, D and E, respectively. In this case, grades A, B, C, D and E, will be awarded in the ratio 1: 3 : 5 : 4 : 3. a Using the examiners’ suggestion, represent the scores in a cumulative frequency polygon and use it to estimate the median score. b All of the grade boundaries are later reduced by 10%. Estimate how many candidates will be awarded a higher grade because of this. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency 13 |
The values of x shown in the following table are to be represented in a bar chart. x Frequency 5 2 6 5 7 9 8 10 9 9 10 5 11 2 a i Sketch a curve that shows the shape of the data. ii Find the mode, mean and the median of x. b The two smallest values of x (i.e. 5 and 5) are changed to 21 and 31. Investigate the effect that this has on the mode, the mean, the median and on the shape of the curve. c If, instead, the two largest values of x (i.e. 11 and 11) are changed to –9 and b, so that the mean of x decreases by 1, find the value of b and investigate the effect that this has on the mode, the median and the shape of the curve. 14 A histogram is drawn to illustrate a set of continuous data whose mean and median are equal. Make sketches of the different types of curve that could be drawn to represent the shape of the histogram. 15 Students’ marks in a Biology examination are shown by percentage in the following table. Marks ( )% Frequency ( )% 20– 5 30– 10 40– 20 50– 30 60– 20 70– 10 80–90 5 a Without drawing an accurate histogram, describe the shape of the set of marks. What does the shape suggest about the values of the mean, the median and the mode? b Information is provided about the marks in examinations in two other subjects: Chemistry: mode > median > mean Physics: mean > median > mode Sketch a curve to show the shape of the distribution of marks in each of these exams. 49 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Chapter 2: Measures of central tendency Checklist of learning and understanding ● Measures of central tendency are the mode, the mean and the median. ● For ungrouped data, the mode is the most frequently occurring value. ● For grouped data, the modal class has the highest frequency density and the greatest height column in a histogram. ● For ungrouped data, x ● For grouped data, x = x n= Σ. xf Σ f Σ or fx Σ |
f Σ. ● The formulae for ungrouped and grouped coded data can be summarised by: x = mean ( – ) mean ( ax b − ) + b ] ● For ungrouped coded data Σ ( ) ax b − n + b ● For grouped coded data 50 x = 1 a Σ ( ) ax b f − f Σ + b ● For ungrouped data, the median is at the n 1 + 2 th value. n 2 ● For grouped data, we estimate the median to be at the th value on a cumulative frequency graph. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency END-OF-CHAPTER REVIEW EXERCISE 2 1 For each of the following sets of data, decide whether you would expect the mean to be less than, equal to or greater than the median and the mode. a The ages of patients receiving long-term care at a hospital. b The numbers of goals scored in football matches. c The heights of adults living in a particular city. 2 The mean mass of 13 textbooks is 875 grams, and n novels have a total mass of 13 706 grams. Find the mean mass of a novel, given that the textbooks and novels together have a mean mass of 716.6 grams. 3 Nine values are 7, 13, 28, 36, 13, 29, 31, 13 and x. a Write down the name and the value of the measure of central tendency that can be found without knowing the value of x. b If it is known that x is greater than 40, which other measure of central tendency can be found and what is its value? c If the remaining measure of central tendency is 25, find the value of x. 4 For the data shown in the following table, x has a mean of 7.15. x Frequency 3 a 6 b 10 15 c d [1] [1] [1] [3] [1] [1] [2] a Find the mean value of y given in |
the following table. [1] 51 y 11 14 18 Frequency a b c 23 d b Find a calculated estimate of the mean value of z given in the following table. z Frequency 2– a 8– b 14– 24–34 c d 5 The table below shows the number of books read last month by a group of children. No. books No. children 2 3 3 8 4 15 5 q a If the mean number of books read is exactly 3.75, find the value of q. b Find the greatest possible value of q if: i the modal number of books read is 4 ii the median number of books read is 4. 6 The following table gives the heights, to the nearest 5cm, of a group of people. Heights (cm) 120–135 140–150 155–160 165–170 175 –185 No. people 30 p 12 16 21 Given that the modal class is 140–150 cm, find the least possible value of p. Copyright Material - Review Only - Not for Redistribution [2] [2] [1] [1] [3] Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Chapter 2: Measures of central tendency 7 The following histogram illustrates the masses, m kilograms, of the 216 sales of hay that a farmer made to customers last year. a Show that a calculated estimate of the mean is equal to the median. [4] b Estimate the price per kilogram at which the hay was sold, given that these sales generated exactly $1944. Why is it possible that none of the customers actually paid this amount per kilogram for the hay? [4] 8 An internet service provider wants to know how customers rate its services. A questionnaire asks customers to tick one of the following boxes.2 5.4 3.6 1.8 0 30 40 50 60 70 80 Mass (m kg) excellent □ good □ average □ poor □ very poor □ a How might the company benefit from knowing each of the available average responses of its customers? [2] b What additional benefit could the company obtain by using the following set of tick boxes instead? 52 excellent = 5 □ good 4 □ = average = 3 □ poor = 2 □ very poor 1 � |
� = [2] 9 The numbers of items returned to the electrical department of a store on each of 100 consecutive days are given in the following table. No. items No. days 0 49 1 16 2 10 3 9 4 7 5 5 p6– 4 a Write down the median. b Is the mode a good value to use as the average in this case? Give a reason for your answer. c Find the value of p, given that a calculated estimate of the mean is 1.5. d Sketch a curve that shows the shape of this set of data, and mark onto it the relative positions of the mode, the mean and the median. [1] [1] [3] [2] 10 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table. Time spent (t hours) 0.1 < < t 0.5 0.6 < < t 1.0 1.1 < < t 2.0 2.1 < <t 3.0 3.1 < < t 4.5 Frequency 11 15 18 30 21 i Draw, on graph paper, a histogram to illustrate this information. [5] ii Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group. [3] Cambridge International AS & A Level Mathematics 9709 Paper 6 Q5 June 2008 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 2: Measures of central tendency [3] [3] [3] [2] [3] [3] [5] 53 11 For 150 values of x, it is given that Σ ( x 1) − + Σ ( x − 4) = 4170. Find x. PS 12 On Monday, a teacher asked eight students to write down a number, which is denoted by x. On Tuesday, when one of these students was absent, they asked them to add 1 to yesterday’s number and write it down. Find the number written down on Monday by the student who was absent on Tuesday, given that x =, and that the mean of Monday’s and |
Tuesday’s numbers combined was 27 1 3. 30 1 4 13 A delivery of 150 boxes, each containing 20 items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table. No. damaged items No. boxes ( )f 0 100 1 10 2 10 3 10 4 10 5 10 6 or more 0 a Find the mode, the mean and the median number of damaged items. b Which of the three measures of central tendency would be the most appropriate to use as the average in this case? Explain why using the other two measures could be misleading. 14 The monthly salaries, w dollars, of 10 women are such that Σ w( − 3000) = − 200. The monthly salaries, m dollars, of 20 men are such that Σ m( − 4000) 120. = a Find the difference between the mean monthly salary of the women and the mean monthly salary of the men. b Find the mean monthly salary of all the women and men together. 15 For 90 values of x and 64 values of y, it is given that Σ − x ( 1) = 72.9 and ( y Σ + 1) = 201.6. Find the mean value of all the values of x and y combined. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 54 Chapter 3 Measures of variation In this chapter you will learn how to: ■ find and use different measures of variation ■ use a cumulative frequency graph to estimate medians, quartiles and percentiles ■ calculate and use the standard deviation of a set of data (including grouped data) either from the data itself or from given totals xΣ and Σx2, or coded totals ( Σ − totals in solving problems that may involve up to two datasets. x b ) and Σ ( x b − )2 and use such Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation PREREQUISITE KNOWLEDGE Where it comes from What you should be able to do Check your skills IGCSE / |
O Level Mathematics Accurately label and read from an axis, using a given scale. Substitute into and manipulate algebraic formulae containing squares and square roots. 1 The numbers 2 and 18 are marked on an axis 20 cm apart. How far apart are the numbers 4.5 and 17.3 on this axis? 2 If = , find the positive value of: a y when =x 13, =a 4 and =b 352 b x when =y 12, =a. 5 and b 11= How do we best summarise a set of data? A measure of central tendency alone does not describe or summarise a set of data fully. Although it may tell us the location of the more central values or the most common values, it tells us nothing about how widely spread out the values are. Two sets of data can have the same mean, median or mode, yet they can be completely different. A better description of a set of data is given by a measure of central tendency and a measure of variation. Variation is also known as spread or dispersion. Consider the runs scored by two batters in their past eight cricket matches, which are given in the following table. Batter A Batter B 25 2 30 70 31 1 26 0 31 43 28 29 24 Total: 224 1 104 3 Total: 224 55 The mean number of runs scored by A and by B is the same; namely, the patterns of the number of runs are clearly very different. The numbers for batter A are quite consistent, whereas the numbers for batter B are quite varied. This consistency (or lack of it) can be indicated by a measure of variation, which shows how spread out a set of data values are. 28. However, 224 8 ÷ = Three commonly used measures of variation are the range, interquartile range and standard deviation. 3.1 The range As you will recall, the range is the numerical difference between the largest and smallest values in a set of data. One advantage of using the range is that it is easy to calculate. However, it does not take the more central values into account but uses only the most extreme values. It is often more informative to state the minimum and maximum values rather than the difference between them. For example, in a test for which the lowest mark is 6 and the highest mark is 19, the range is 19 – 6 13. = For grouped data, we can find a minimum and maximum possible range, using the lower and upper boundary |
values of the data. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 WORKED EXAMPLE 3.1 To the nearest centimetre, the tallest and shortest pupils in a class are 169cm and 150 cm. Find the least and greatest possible range of the students’ heights. Answer Least possible range 168.5 – 150.5 = Greatest possible range 169.5 – 149.5 = 18cm = The intervals in which the given heights, h, lie are 168.5 150.5cm 169.5cm 149.5 and ø < h ø < h. = 20 cm 3.2 The interquartile range and percentiles The lower quartile, median and upper quartile, as you will recall, divide the values in a dataset into four parts, with an equal number of values in each part. These three measures are commonly abbreviated by: ● Q1 for the lower quartile ● Q2 for the median (or middle quartile) ● Q3 for the upper quartile. The interquartile range is the numerical difference between the upper quartile and the lower quartile, and gives the range of the middle half (50%) of the values, as shown in the following diagram. 56 interquartile range smallest value Q1 Q2 Q3 largest value KEY POINT 3.1 Interquartile range upper quartile – lower quartile = or IQR 1. –3 = Q Q The interquartile range is often preferred to the range because it gives a measure of how varied the more central values are. It is relatively unaffected by extreme values, also called outliers, and can be found even when the exact values of these are not known. TIP Ungrouped data The positions of the lower and upper quartiles depend on whether there are an odd or even number of values in the set of data. One method that we can use to find the quartiles is as follows. For an even number of ordered values: we split the data into a lower half and an upper half. Then Q1 and Q3 are the medians of the lower half and upper half, respectively. For an odd number of ordered values: we split the data into a lower |
half and an upper half at the median, which we then discard. Again, Q1 and Q3 are the medians of the lower half and upper half, respectively. Copyright Material - Review Only - Not for Redistribution th always at the In a set of ungrouped data, the median is n 1 + 2 value. However, it is advisable to find the quartiles by inspection rather than by memorising formulae for their positions. Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation WORKED EXAMPLE 3.2 Find the interquartile range of the eight ordered values 2, 5, 9, 13, 29, 33, 49 and 55. Answer 1st 2 2nd 5 3rd 9 4th 13 5th 29 6th 33 7th 49 8th 55 Q1 Q2 Q3 The ordered values are shown, with their positions indicated. IQR = = = = Q Q − 1 3 33 49 + 2 41 7 − 34 − 5 9 + 2 WORKED EXAMPLE 3.3 Find the interquartile range of the seven values 69, 17, 43, 6, 73, 77 and 39. Answer 1st 6 IQR 2nd 17 Q1 3rd 39 4th 43 Q2 5th 69 6th 73 Q3 7th 77 The ordered values and their positions are shown. 57 = = = 3 – Q Q 1 73 – 17 56 REWIND We studied stemand-leaf diagrams in Chapter 1, Section 1.2. WORKED EXAMPLE 3.4 Find the interquartile range of the 13 grouped values shown in the following stem-and-leaf diagram. 14 15 16 Key: 14 2 represents 142 Answer Q2 is at the 13 1 + 2 which is 153. th 7th value, = Q 1 = Q 3 = 144 148 + 2 157 159 + 2 − IQR 158 146 = = 146 = 158 = 12 We identify the median as 153, which we now discard. This leaves a lower half (142 to 151) and an upper half (155 to 168), with six values in each. The median of a group of six values is the 3.5 |
th value. These are marked by red dots in the stemand-leaf diagram. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 EXPLORE 3.1 In this activity, you will investigate the value of the median in relation to the smallest and largest values in a set of data, and also in relation to the lower and upper quartiles. For each ordered set of data, A to D, write down these five values: the smallest value; the lower quartile; the median; the upper quartile; and the largest value. Set A: 2, 2, 3, 11, 11, 21, 22. Set B: 6, 6, 6, 11, 13, 17, 19, 20. Set C: 9, 15, 28, 32, 35, 49. Set D: 5, 7, 9, 10, 11, 12, 12, 16, 17. It may be useful to mark the five values for each dataset on a number line. Use your results to decide which of the following statements are always, sometimes or never true. 1 The median is mid-way between the smallest and largest values. 2 The median is mid-way between the lower and upper quartiles. 3 The interquartile range is equal to exactly half of the range. 4 58 REWIND We estimated the median from a cumulative frequency graph in Chapter 1, Section 1.4. Grouped data We can use a cumulative frequency graph to estimate values in any position in a set of data. This includes the lower quartile, the upper quartile and any chosen percentile. KEY POINT 3.2 For grouped data with total frequency n following table. f= Σ, the positions of the quartiles are shown in the Quartile lower ( )1Q median ( )2Q upper ( )3Q Position n 4 or 1 4 f Σ n 2 or 1 2 Σ f n3 4 or 3 4 Σ f The nth percentile is the value that is n% of the way through a set of data. Q Q,1 2 and Q3 are the 25th, 50th and 75th percentiles, respectively. In an ordered dataset with, say, 320 values, Q Q, |
1 240th values, and the 90th percentile is at the 2 and Q3 are at the 80th, 160th and 288th value. (0.90 320) × = The range of the middle 80% of a dataset is the difference between the 10th and 90th percentiles. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation WORKED EXAMPLE 3.5 The following graph illustrates the times, in minutes, taken by 500 people to complete a task. Use the graph to find an estimate of: a the greatest possible range b the interquartile range c the 95th percentile 500 475 400 375 300 250 200 125 100 0 59 5 10 Q1 15 Q3 Time (min) 20 25 30 95th percentile Answer a 30 – 2 = 28 min The greatest possible range is equal to the width of the polygon. b Q 1 Q 3 IQR ≈ ≈ = ≈ = 8.0 min 14.5 min 3 – Q Q 1 14.5 – 8.0 6.5 min We locate the quartiles, then estimate their values by reading from the graph. Lower quartile: Upper quartile: = n 4 3 n 4 = 500 4 3 500 × 4 = 125 th value = 375 th value c ≈ 24.0 min The 95th percentile is at the (0.95 500) × = 475th value. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Box-and-whisker diagrams A box-and-whisker diagram (or box plot) is a graphical representation of data, showing some of its key features. These features are its smallest and largest values, its lower and upper quartiles, and its median. If drawn by hand, the diagram is best drawn on graph paper and must include a scale. It takes the form shown in the following diagram, which shows some features of a dataset denoted by x. 0 5 10 15 x smallest value Q1 |
Q2 box whisker Q3 largest value Key features of the data for x represented in the box-and-whisker diagram are: Median: Range 14 – 1 13 = – IQR Q Q 1 =Q 6 2 = = = 3 11 – 4 = 7 The following box-and-whisker diagram is a representation of a dataset denoted by y, drawn using the same scale as the previous diagram. 0 5 10 15 y 60 The following table shows a measure of central tendency and two measures of variation for each of x and y, and we can use these to make comparisons. Median Range IQR Dataset x Dataset y 6 9 13 7 15 – 0 15 = 12 – 3 9 = By comparing medians, values of x are, on average, less than values of y. By comparing ranges and interquartile ranges, values of y are more varied than values of x. We can assess the skewness of a set of data using the quartiles in a box-and-whisker diagram. In the previous box-and-whisker diagram for x, 1, and the longer tail of the curve drawn over a bar chart would be to the side of the larger values. This means that the data for x is positively skewed In the previous box-and-whisker diagram for y, 2, and the longer tail of the curve would be to the side of the smaller values. This means that the data for y is negatively skewed reasonably symmetrical set of data would have 1 Copyright Material - Review Only - Not for Redistribution TIP The whisker (which shows the range) is not drawn through the box (which shows the interquartile range). Items and, where appropriate, units such as ‘Length (cm)’ and ‘Mass (kg)’ must be indicated on the diagram. REWIND We looked briefly at positively and negatively skewed sets of data in Chapter 2, Section 2.3. Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation EXERCISE 3A 1 Find the range and the interquartile range of the following sets of data. a 5, 8, 13, 17, 22, 25, 30 b 7, 13, 21, 2, 37, 28, 17, 11, |
2 c 42, 47, 39, 51, 73, 18, 83, 29, 41, 64 d 113, 97, 36, 81, 49, 41, 20, 66, 28, 32, 17, 107 e 4.6, 0, –2.6, 0.8, –1.9, –3.3, 5.2, –3.2 2 a Find the range and the interquartile range of the dataset represented in the following box plot. 0 1 2 3 4 b What type of skewness would you expect this set of data to have? 3 The following stem-and-leaf diagram shows the marks out of 50 obtained by 15 students in a Science test Key: 2 5 represents a mark of 25 out of 50 61 a Find the range and interquartile range of the marks. b Illustrate the data in a box-and-whisker diagram on graph paper and include a scale. c For this set of data, express Q3 in terms of Q1 and Q2. 4 The numbers of fouls made in eight hockey matches and in eight football matches played at the weekend are shown in the following back-to-back stem-and-leaf diagram. Hockey (8) Football (8 Key: 6 1 8 represents 16 fouls in hockey and 18 fouls in football a Is it true to say that the numbers of fouls in the two sports are equally varied? Explain your answer. b Draw two box-and-whisker diagrams using the same scales. Write a sentence to compare the numbers of fouls committed in the two sports. 5 Rishi and Daisy take the same seven tests in Mathematics, and both students’ marks improve on successive tests. Their percentage marks are as follows. Rishi’s marks Daisy’s marks 15 24 28 33 39 42 50 51 65 69 72 78 83 86 a Explain why it would not be useful to use the range or the interquartile range alone as measures for comparing the marks of the two students. b Name two measures that could be used together to give a meaningful comparison of the two students’ marks. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 6 The following table |
shows the maximum speeds, s km/h, of Maximum speed No. vintage some vintage cars. s( km/ h) cars f( ) a On the same sheet of graph paper, construct a cumulative frequency polygon and a box-and-whisker diagram to illustrate the data. b Use your box-and-whisker diagram to assess what type of skewness the data have. 7 Twenty adults are selected at random, and each is asked to state the number of trips abroad that they have made. The results are shown in the following back-to-back stem-and-leaf diagram. <s 35 s <<< s <<< s <<< s <<< s <<< s <<< 40 45 50 55 70 75 35 40 45 50 55 70 0 20 65 110 27 13 5 a i Draw box-and-whisker diagrams, using the same scales for males and for females. ii Interpret the key features of the data represented in your diagrams and compare the data for the two groups of adults. b In a summary of the data, a student writes, ‘The females have visited more countries than the males.’ Is this statement justified? Give a reason to support your answer. Males (9) Females (11 Key: 3 1 1 represents 13 trips for a male and 11 trips for a female 8 The resistances, in ohms (Ω, of 100 conductors are represented in the following graph. ) 62 ) 100 80 60 40 20 0 0.1 0.2 0.3 0.4 0.5 0.6 Resistance (Ω) Find, to an appropriate degree of accuracy, an estimate of: a c the interquartile range the percentile that is equal to 0.192 Ω b d the 90th percentile the range of the middle 40% of the resistances. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 9 The areas, in cm2, of some circuit boards are represented in the following graph. Chapter 3: Measures of variation ( 200 150 100 50 0 8 16 24 32 40 48 56 Area (cm2) a State the greatest possible range of the data. b Construct a box-and-whisker diagram to illustrate the data. c Find the range of |
the middle 60% of the areas. d An outlier is an extreme value that is more than 1.5 times the interquartile range above the upper quartile or more than 1.5 times the interquartile range below the lower quartile. Find the areas that define the outliers in this set of data, and estimate how many there are. How accurate is your answer? 63 10 A company manufactures right-angled brackets for use in the construction industry. A sample of brackets are measured, and the number of degrees by which their angles deviate from a right angle are summarised in the following table. a Draw a cumulative frequency polygon to illustrate these deviations. b Estimate the median and the interquartile range of the bracket angles, giving both answers correct to 1 decimal place. c A bracket is considered unsuitable for use if its angle deviates from a right angle by more than 1.2°. Estimate what percentage of this sample is unsuitable for use, giving your answer correct to the nearest integer. Deviation from 90° d( ) No. brackets f( ) <d –1.5 –1.5 ø < d –1.0 –1.0 ø < d –0.5 –0.5 ø < d 0.0 0.0 ø < d 0.5 0.5 ø < d 1.0 1.0 ø < d 1.5 1.5 ø < d 2.5 0 24 46 61 34 34 20 17 11 The following table shows the cumulative frequencies for values of x. x cf < 0 0 < 10 12 < 15 30 < 25 90 < 30 102 < 40 120 Without drawing a cumulative frequency graph, find: a b the interquartile range the 85th percentile. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 M 12 Fifty 10-gram samples of a particular type of mushroom are collected by volunteers at a university and tested. The following table shows the mass of toxins, in hundredths of a gram, in these samples. Mass (/0.01g) No. 10 g samples 0– 2 4– 19 11– 23 17– 3 20–30 3 a Draw a cumulative |
frequency curve to illustrate the data. b Use your curve to estimate, correct to 2 decimal places: i the interquartile range ii the range of the middle 80%. c It was found that toxins made up between 0.75% and 2.25% of the mass of n of these samples. Use your curve to estimate the value of n. d Make an assessment of the variation in the percentage of toxic material in these samples. Can you suggest any possible reasons for such variation? M 13 A 9-year study was carried out on the pollutants released when biomass fuels are used for cooking. Researchers offered nearly 1000 people living in 12 villages in southern China access to clean biogas and to improved kitchen ventilation. Some people took advantage of neither; some changed to clean fuels; some improved their kitchen ventilation; and some did both. The following diagram shows data on the concentrations of nitrogen dioxide in these people’s homes at the end of the study. 64 Groups: Neither Clean fuels only Ventilation only Both Nitrogen dioxide pollutant concentration (mg/m3) 0 0.25 0.5 0.75 Study the data represented in the diagram and then write a brief analysis that summarises the results of this part of the study. DID YOU KNOW? The study of human physical growth, auxology, is a multidisciplinary science involving genetics, health sciences, sociology and economics, among others. Exceptional height variation in populations that share a genetic background and environmental factors is sometimes due to dwarfism or gigantism, which are medical conditions caused by specific genes or abnormalities in the production of hormones. In regions of poverty or warfare, environmental factors, such as chronic malnutrition during childhood, may result in delayed growth and/or significant reductions in adult stature even without the presence of these medical conditions. At the time of their meeting in London in 2014, Chandra Bahadur Dangi (at 54.6 cm) and Sultan Kosen (at 254.3 cm) were the shortest and tallest adults in the world. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation EXPLORE 3.2 The interquartile range is based around the median. In this exploration, we investigate a possible way to define variation based on the mean. Choose a set of five |
numbers with a mean of 10. The deviation of a number tells us how far and to which side of the mean it is. Numbers greater than the mean have a positive deviation, whereas numbers less than the mean have a negative deviation, as indicated in the following diagram. mean negative deviation positive deviation Find the deviation of each of your five numbers and then calculate the mean deviation. Compare and discuss your results, and investigate other sets of numbers. Can you predict what the result will be for any set of five numbers with a mean of 10? Can you justify your prediction? What would you expect to happen if you started with any set of five numbers? 3.3 Variance and standard deviation In the Explore 3.2 activity, you discovered that the mean deviation is not a useful way of measuring the variation of a dataset because the positive and negative deviations cancel each other out. So, if we want a measure of variation around the mean, we need to ensure that each deviation is positive or zero. We can do this by calculating the mean distance of the data values from the mean, which TIP means we x x− calculate −x x and remove the minus sign if the answer is negative. 65 we call the ‘mean absolute deviation from the mean’, x x Σ − n. However, it is hard to calculate this accurately or efficiently for large sets of data and it is difficult to work with algebraically, so this approach is not used in practice. ) 2 for all data values and find Alternatively, we can calculate the squared deviation, their mean. This is the ‘mean squared deviation from the mean’, which we call the variance of the data. ( −x x Σ 2 ) ( Var( x ) = x x − n For measurements and deviations in metres, say, the variance is in m2. So, to get a measure of variation that is also in metres, we take the square root of the variance, which we call the standard deviation. Standard deviation of x = Var( x ) = )2 ( x x Σ − n This looks no easier to calculate than the ‘mean absolute deviation from the mean’, however, the formula for variance can be simplified (see appendix at the end of this chapter) to give: Var n We can find the variance and standard deviation from n, Σx and Σx2, which are the number of values, their sum and the sum of |
their squares, respectively. We often use the abbreviation ) to represent the standard deviation of X. XSD( Copyright Material - Review Only - Not for Redistribution TIP To find Σx2, we add up the squares of the data values. A common error is to add up the data values and then square the answer but this would be written as instead. 2x) (Σ Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 TIP To find each value of x f2 multiply x by xf. If we multiply x by f and then square the answer, we will obtain which is not required. (which is the same as fx2), we can either multiply x2 by f or we can 2 x f xf )2 = ( 2, A low standard deviation indicates that most values are close to the mean, whereas a high standard deviation indicates that the values are widely spread out from the mean. Consider a drinks machine that is supposed to dispense 400 ml of coffee per cup. We would expect some variation in the amount dispensed, yet if the standard deviation is high then some customers are likely to feel cheated and some risk being injured because of their overflowing cups! KEY POINT 3.3 For ungrouped data: Standard deviation = Variance = ), where x = x Σ n. For grouped data: Standard deviation = Variance = (, where x = xf Σ f Σ. TIP We can remember the formula for variance as ‘mean of the squares minus square of the mean’. 66 WORKED EXAMPLE 3.6 For the set of five numbers 3, 9, 15, 24 and 29, find: a the standard deviation b which of the five numbers are more than one standard deviation from the mean. Answer a Variance = 2 2 x Σ n − x Σ + 15 + 2 24 2 29 + 5 2 80 5 2 1732 5 − 346.4 16 − = = = = Standard deviation 90.4 = = 90.4 9.51 b 16 9.51 6.49 = − 16 9.51 25.51 = + − + + 3 9 |
15 24 29 + 5 + We subtract the square of the mean from the mean of the squares to find the variance. 2 We take the square root of the variance to find the standard deviation, correct to 3 significant figures. We find the values that are 9.51 below and 9.51 above, using the mean of 16. The numbers are 3 and 29. Identify which of the five numbers are outside the range 6.49 number < <. 25.51 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation WORKED EXAMPLE 3.7 Find the standard deviation of the values of x given in the following table, correct to 3 significant figures. x 12 14 16 Answer x 12 14 16 f 13 28 10 f 13 28 10 xf 156 392 160 2 x f ××= x xf 12 156 1872 = × 14 392 5488 × = 16 160 × = 2560 Σ =f 51 Σ =xf 708 2x fΣ = 9920 SD 9920 51 − 708 51 = 1.34 What happens if we use a rounded value for the mean? The table shown opposite is an extended frequency table that is used to find fΣ, Σxf and x f2Σ, which are needed to calculate the standard deviation. We use the totals 51, 708 and 9920 to find the standard deviation. TIP Note how the values of x f2 are calculated. TIP Always use the exact value of the mean to calculate variance and standard deviation. 67 Correct to 1 decimal place, the mean in Worked example 3.7 is 708 ÷ 51 13.9. = If we use x 13.9 = error of 0.2. in our calculation, we obtain SD( x ) = 9920 51 − 13.9 2 = 1.14. This is an The rounded mean has caused a substantial error (0.2 is about 15% of the correct value 1.34). So, when calculating the variance or standard deviation, always use xf Σ f Σ, rather than a rounded value for the mean. x Σ or n When data are grouped, |
actual values cannot be seen, but we can calculate estimates of the variance and standard deviation. The formulae in Key point 3.3 are used to do this, where x now represents class mid-values and x = xf Σ f Σ is a calculated estimate of the mean. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 WORKED EXAMPLE 3.8 Calculate an estimate of the standard deviation of the heights of the 20 children given in the following table. Height (metres) No. children f( ) Answer 1.2– 2 1.4– 12 1.5–1.7 6 We extend the frequency table to include class mid-values x( find the totals Σf, Σxf and x f2Σ shown in the table opposite. ), and to, as Height (m) 1.2– 1.4– 1.5–1.7 No. children f( ) Mid-value x( ) xf x f2 2 1.3 2.6 12 1.45 17.4 6 1.6 9.6 fΣ = 20 xfΣ = 29.6 3.38 25.23 15.36 2x fΣ = 43.97 Estimate of standard deviation 43.97 20 − 29.6 20 0.0081 0.09 m = = = = 68 EXPLORE 3.3 Four students analysed data that they had collected. Their findings are given below. 1 Property prices in a certain area of town have a high standard deviation. 2 The variance of the monthly sales of a particular product last year was high. 3 The standard deviation of students’ marks in a particular examination was close to zero. 4 The times taken to perform a new medical procedure have a low variance. Discuss the students’ findings and give a possible description of each of the following. 1 The type of environment and the people purchasing property in this area of town. 2 The type of product being sold. 3 The usefulness of the examination. 4 The efficiency of the teams performing the medical procedures. Copyright Material - Review Only - Not for Redistribution Review |
Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation Although standard deviation is far more commonly used as a measure of variation than the interquartile range, it may not always be ideal because it can be significantly affected by extreme values. The interquartile range may be better, and a box-and-whisker diagram is often much more useful as a visual representation of data than the mean and standard deviation. Some features of the standard deviation are compared to the interquartile range in the following table. Advantages of standard deviation Disadvantages of standard deviation Much simpler to calculate than the IQR. Far more affected by extreme values than the IQR. Data values do not have to be ordered. Gives greater emphasis to large deviations than to small deviations. Easier to work with algebraically when doing more advanced work. Takes account of all data values. EXERCISE 3B 1 Find the mean and the standard deviation for these sets of numbers. a 27, 43, 29, 34, 53, 37, 19 and 58. b 6.2, 8.5, 7.7, 4.3, 13.5 and −11.9. − − 2 Last term Abraham sat three tests in each of his science subjects. His raw percentage marks for the tests, in the order they were completed, are listed. Biology Chemistry Physics 21 33 45 41 53 65 51 63 75 69 a Calculate the variance of Abraham’s marks in each of the three subjects. b Comment on the three values obtained in part a. Do the same comments apply to Abraham’s mean mark for the tests in the three subjects? Justify your answer. 3 The following table shows the number of pets owned by each of 35 families. No. pets No. families f( ) 0 6 1 12 2 9 3 4 4 3 5 1 Find the mean and variance of the number of pets. 4 The numbers of cobs produced by 360 maize plants are shown in the following table. No. cobs No. plants f( ) 0 11 1 75 2 185 3 81 4 8 a Calculate the mean and the standard deviation. b Find the interquartile range and give an example of what it tells us about this dataset that the standard deviation does not tell us. Copyright Material - Review Only - Not for Red |
istribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 5 The times spent, in minutes, by 30 girls and by 40 boys on an assignment are detailed in the following table. Time spent (min) No. girls f( ) No. boys f( ) 20– 6 15 30– 14 11 40– 60–80 7 7 3 7 a For the boys and for the girls, calculate estimates of the mean and standard deviation. b It is required to make a comparison between the times spent by the two groups. i What do the means tell us about the times spent? ii Use the standard deviations to compare the times spent by the two groups. 6 The lengths, correct to the nearest centimetre, of 50 rods are given in the following table. Length (cm) 15–17 18–24 25–29 30–37 No. rods f( ) 13 18 11 8 Calculate an estimate of the standard deviation of the lengths. 7 For the dataset denoted by x in the following table, k is a constant. x f 15 2k 16 17 5+k – 3k 18 10 19 8 20 3 70 Find the value of k and calculate the variance of x, given that 17=x. 8 The following table illustrates the heights, in centimetres, of 150 children. Height (cm) No. children f( ) 140 up to 144 144 up to 150 150 up to 160 160 up to 165 a b 69 28 a Given that a calculated estimate of the mean height is exactly 153.14 cm, show that 142 a + 147 b = 7726, and evaluate a and b. b Calculate an estimate of the standard deviation of the heights. PS 9 Kristina plans to raise money for charity. Her plan is to walk 217 km in 7 days so that she walks km2 2 – n n +k compare this with the interquartile range. on the nth day. Find the standard deviation of the daily distances she plans to walk, and PS 10 The mass of waste produced by a school during its three 13-week terms is given in tonnes, correct to 2 decimal places, in the following table. Mass of waste (tonnes) 0.15– 0.29 0.30– 0.86 0.87– |
1.35 1.36–2.00 No. weeks f( ) 5 8 20 6 a Calculate estimates of the mean and standard deviation of the mass of waste produced per week, giving both answers correct to 2 decimal places b No waste is produced in the 13 weeks of the year that the school is closed. If this additional data is included in the calculations, what effect does it have on the mean and on the standard deviation? Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation PS 11 The ages, in whole numbers of years, of a hotel’s 50 staff are given in the following table. Calculated estimates of the mean and variance are 37.32 and 69.1176, respectively. Age (years) No. staff f( ) 23–30 14 31–37 x 38– 45 46–59 y 6 Exactly 1 year after these calculations were made, Gudrun became the 51st staff member and the mean age became exactly 38 years. Find Gudrun’s age on the day of her recruitment, and determine what effect this had on the variance of the staff’s ages. What assumptions must be made to justify your answers? PS P 12 Refer to the following diagram. In position 1, a 10-metre rod is placed 10 metres from a fixed point, P. Six small discs, A to F, are evenly spaced along the length of the rod. The rod is rotated anti-clockwise about its centre by ° to position 2. The distances from P to the discs are denoted by x. 30α = P 10 Position 1 Position 2 A B C D E F 10 P 10 A B 10 C D E F a What effect does the 30° rotation have on values of x? Investigate this by first considering the effect on the average distance from P to the discs. b Find two values that can be used as measures of the change in the variation of x caused by the rotation. 71 c Use the values obtained in parts a and b to summarise the changes in the distances from P to the discs caused by the rotation. 2Σx is constant for all values of α? (Hint: to do this, you need only to show that d Can you prove that constant |
for 0 < 90α° < °). 2Σx is EXPLORE 3.4 Twenty adults completed as many laps of a running track as they could manage in 30 minutes. The following table shows how many laps they completed. Completed laps No. adults f( ) 4–8 6 9–13 10 14–18 4 Two students, Andrea and Billie, were asked to calculate an estimate of the standard deviation. Their working and answers, which you should check carefully, are shown below. Andrea: ( Billie: 2 6 6 × ( ) + 2 ( 11 10 × 20 ) + 2 ( 16 4 × ) 2 × 2 6.5 6 11.5 10 16.5 4 × 20 ) + ) + × ( ( 2 − ) 2 10.5 3.5 = 2 11 − 3.5 = Compare Andrea and Billie’s approaches. What have they done differently and why do you think they did so? Is one of their answers better than the other? If so, in what way? Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Calculating from totals For ungrouped data, we calculate variance (Var) and standard deviation (SD) from totals n, Σx and 2Σx. For grouped data, we calculate using totals Σf, Σxf and 2Σx f. In both cases, we can rearrange the formula for variance if we wish to evaluate one of the totals. WORKED EXAMPLE 3.9 Given that, 25=n Σ =x 275 and Var( 7=x ), find 2Σx. Answer 2 2 x Σ 25 − 275 25 = 7 2 Σ x = = × 25 3200 7 + ( 275 25 2 ) Substitute the given values into the formula for variance, then rearrange the terms to make x2Σ the subject. 72 Combined sets of data In Chapter 2, Section 2.2, sets of data were combined by simply considering all of their values together, and we learned how to find |
the mean. Here, we consider the variation of datasets that have been combined in the same way. The variance and standard deviation of a combined dataset are calculated from its totals, which are the sums of the totals of the two sets from which it has been made. 4 and 2 3 The two sets {1, 2, 3, 4} and {4, 5, 6} individually have variances of 1 1 set {1, 2, 3, 4, 4, 5, 6} has a variance of approximately 2.53.. The combined WORKED EXAMPLE 3.10 The heights, cmx, of 10 boys are summarised by Σ =x 1650 The heights, cmy, of 15 girls are summarised by Σ =y 2370 = 2Σ and x 2Σ =y and 275 490.. 377835 Calculate, to 3 significant figures, the standard deviation of the heights of all 25 children together. Answer 2 Σ x 2 y + Σ = 275490 377835 + = 653325 Σ + Σ = x y 1650 2370 + = 4020 For the 25 children, we find the sum of the squares of their heights and the sum of their heights. Standard deviation 653 325 25 16.6cm = = 2 − 4020 25 We substitute the three sums into the formula for standard deviation and evaluate this to the required degree of accuracy. TIP The variance of two combined datasets x and y is not (in general) equal to Var( x In Worked example 3.10, Var(boys) 324 = and Var(girls) 225 but Var(boys and girls) = ≠ ) + Var( y ) 2. 324 225 + 2. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation If two sets of data, denoted by x and y, have nx and ny values, respectively, then the mean y) (Σ + Σ and and variance of their combined values are found using the totals n x (). KEY POINT 3.4 The mean of x and y combined is. The variance of x and y combined is � |
�. We can rearrange the formulae in Key point 3.4 if we wish to find one of the totals involved. WORKED EXAMPLE 3.11 In an examination, the percentage marks of the 120 boys are denoted by x, and the percentage marks of the 80 girls are denoted by y. The marks are summarised by the totals x =Σ, 7020 2xΣ = 424 320 and 2yΣ =. 352 130 Calculate the girls’ mean mark, given that the standard deviation for all these students is 10. Answer 424320 352130 + 120 80 + − 7020 y + Σ 120 80 + 3882.25 − ( 7020 y + Σ 40 000 2 2 ) 2 10 = = 100 We substitute the given values into the formula for variance, knowing that this is equal to102. Then multiply throughout by 40 000. 73 155 290 000 − 7020 7020 ( ( = 4000 000 = = = 151290 000 151290 000 − 7020 5280 Then we rearrange to make yΣ (the total marks for the girls) the subject. We take the positive square root, as we know that all of the y values are non-negative. Girls’ mean mark = 5280 80 = 66 Divide the total marks for the girls by the number of girls. EXERCISE 3C 1 Given that: a b c d e 2Σ = v 5480, v Σ = 288 and n 64=, find the variance of v. 2Σ w 2Σ x f = 4000, w 5.2= and n 36=, find the standard deviation of w. = 6120, f Σ = 40 and the standard deviation of x is 12, find xfΣ. Σ xf = 2800, f Σ = 50 and the variance of x is 100, find x f2Σ. 193144 2Σ =, t number of data values of t. 2324 t Σ = and that the standard deviation of t is 3, find the Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy |
Cambridge International AS & A Level Mathematics: Probability & Statistics 1 2 A building is occupied by n companies. The number of people employed by these companies is denoted by x. Find the mean number of employees, given that 2Σ and that the standard deviation of x is 18. x, x 220 8900 Σ = = 2Σ = 3 Twenty-five values of p are such that p 2Σ q are such that q 387 = of p and q together. and q 6114 Σ = 6006 and p, and 25 values of. Calculate the variance of the 50 values Σ = 388 4 In a class of 30 students, the mean mass of the 14 boys is 63.5kg and the mean mass of the girls is 57.3kg. Calculate the mean and standard deviation of the masses of all the students together, given that the sums of the squares of the masses of the boys and girls are 58444kg2 and 56222 kg2, respectively. 5 The following table shows the front tyre pressure, in psi, of five 4-wheeled vehicles, A to E. Front-left tyre Front-right tyre A 26 24 B 29 27 C 30 31 D 34 30 E 26 28 a Show that the variance of the pressure in all of these front tyres is 7.65psi2. 2Σ b Rear tyre pressures for these five vehicles are denoted by x. Given that x 7946 and that the variance of the pressures in all of the front and rear tyres on these five vehicles together is 31.6275psi2, find the mean pressure in all the rear tyres. = 74 2Σ 6 The totals x = 7931, x 397 Σ = and y Σ = 499 are given by 29 values of x and n values of y. All the values of x and y together have a variance of 52. a Express y2Σ in terms of n. b Find the value of n for which y 2 Σ − Σ x 2 =. 10 PS 7 The five values in a dataset have a sum of 250 and standard deviation of 15. A sixth value is added to the dataset, such that the mean is now 40. Find the variance of the six values in the dataset. PS 8 A group of 10 friends played a mini-golf competition. Eight of the friends tied for second place, each with a score of 34, and the other two friends tied for first place. Find the winning score |
, given that the standard deviation of the scores of all 10 friends was 1.2 and that the lowest score in golf wins. PSM 9 An author has written 15 children’s books. The first eight books that she wrote contained between 240 and 250 pages each. The next six books contained between 180 and 190 pages each. Correct to 1 decimal place, the standard deviation of the number of pages in the 15 books together is 31.2. Show that it is not possible to determine a specific calculated estimate of the number of pages in the author’s 15th book. PS 10 A set of n pieces of data has mean x and standard deviation S. Another set of n2 pieces of data has mean x and standard deviation 1 2 S. Find the standard deviation of all these pieces of data together in terms of S. FAST FORWARD We will see how standard deviation and probabilities are linked in the normal distribution in Chapter 8, Section 8.2. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation EXPLORE 3.5 The following table shows three students’ marks out of 20 in the same five tests. Amber Buti Chen 1st 12 11 15 2nd 17 16 20 3rd 11 10 14 4th 9 8 12 5th 16 15 19 x x – 1 x 3+ Note that Buti’s marks are consistently 1 less than Amber’s and that Chen’s marks are consistently 3 more than Amber’s. This is indicated in the last column of the table. For each student, calculate the variance and standard deviation. Can you explain your results, and do they apply equally to the range and interquartile range? Coded data What effect does addition of a constant to all the values in a dataset have on its variation? And how can we find the variance and standard deviation of the original data from the coded data? In the Explore 3.5 activity, you discovered that the datasets x, x – 1 and x 3+ have identical measures of variation. The effect of adding –1 or 3+ is to translate the whole set of values, which has no effect on the pattern of spread, as shown in the following diagram. The marks of the three students have the same variance and the same |
standard deviation. 8 9 10 11 12 13 14 15 16 17 18 19 20 REWIND We saw in Chapter 2, Section 2.2 that the mean of a set of data can be found from a coded total such as Σ −. ) x b ( 75 Amber Buti Chen KEY POINT 3.5 For ungrouped data − ) ( x b Σ − n 2 Σ 2 x n For grouped data − ) ( x b f Σ − f Σ 2 These formulae can be summarised by writing Var( ) Var( – ) x b. x = Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 For two datasets coded as (x − a) and (x − b), we can use the coded totals x a) 2xΣ, yΣ and y b)2 Σ( − and y b) the combined set of values of x and y., x a)2 Σ( − 2yΣ, from which we can find the variance of to find xΣ, Σ( − Σ( −, WORKED EXAMPLE 3.12 Eight values of x are summarised by the totals ( xΣ − 2 10) = 1490 and ( xΣ − 10) 100 =. Twelve values of y are summarised by the totals ( + 5) y Σ 2 = 5139 and ( + 5) Σ y = 234. Find the variance of the 20 values of x and y together. Answer 100 8 x = + 10 = 22.5, so Σ = × x 8 22.5 180. = Var(x) = Var(x − 10) = 2 1490 8 − 100 8 = 30. 2 x Σ 8 − 22.5 2 = 30, so Σ 2 =x 4290. y = 234 12 − = 5 14.5, so Σ = y 12 14.5 174. × = We |
find the totals xΣ and 2xΣ. We find the totals yΣ and 2yΣ. 76 2 5139 12 − 234 12 = 48. Var(y) = Var(y + 5) = y Σ 12, so Σ 14.5 48 = − 2 2 2 =y Var(x and y + 12. 3099 12 + = 4290 + 3099 20 − ( 180 + 174 20 2 ) = 56.16 WORKED EXAMPLE 3.13 It is known that 20 girls each have at least one brother. The number of brothers that they have is denoted by x. Information about the values of x – 1 is given in the following table. x – 1 No. girls f Use the coded values to calculate the standard deviation of the number of brothers, to 3 decimal places. Answer x – 1 No. girls f( ) ( x – 1) f ( x – 1) 16 32 3 5 15 45 4 1 4 16 Σ =f 20 ( x Σ − 1) f = 39 ( x Σ − 1) 2 f = 97 We extend the frequency table to find the necessary totals. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation We know that the standard deviation of x and the standard deviation of x( – 1) are equal. SD( ) SD( = x x − 1) 1) f ( x Σ − f Σ 2 − 2 = 2 f ( 1) x Σ − f Σ − 39 20 97 20 1.023 = = EXERCISE 3D 1 Two years ago, the standard deviation of the masses of a group of men and a group of women were 8kg and 6kg, respectively. Today, all the men are 5kg heavier and all the women are 3kg lighter. Find the standard deviation for each group today. 2 Twenty readings of y are summarised by the totals y( Σ − 5) 2 = 890 and y( Σ − |
5) 130 =. Find the standard deviation of y. 3 The amounts of rainfall, r mm, at a certain location were recorded on 365 consecutive days and are summarised by Calculate the mean daily rainfall and the value of r2Σ. r( Σ − 3) = 2 9950 and r( Σ −. 3) 1795.8 = 4 Exactly 20 years ago, the mean age of a group of boys was 15.7 years and the sum of the squares of their ages was 16000. If the sum of the squares of their ages has increased by 8224 in this 20-year period, find the number of boys in the group. 77 5 Readings from a device, denoted by y, are such that, y 105 Σ = the standard deviation of y is 13. Find the number of readings that were taken. y( Σ − 2775 3) = 2 and M 6 Mei measured the heights of her classmates and, after correctly analysing her data, she found the mean and standard deviation to be 163.8cm and 7.6cm. Decide whether or not these measures are valid, given the fact that Mei measured all the heights from the end of the tape measure, which is exactly 1.2 cm from the zero mark. Explain your answers. M 7 A transport company runs 21 coaches between two cities every week. In the past, the mean and variance of the journey times were 4 hours 35 minutes and 53.29 minutes2. What would be the mean and standard deviation of the times if all the coaches departed 10 minutes later and arrived 5 minutes earlier than in the past? Are there any situations in which achieving this might actually be possible? PS 8 During a sale, a boy bought six pairs of jeans, each with leg length x cm. He also bought four pairs of pants, each with leg length x( – 2) cm. The boy is quite short, so his father removed 4cm from the length of each trouser leg. Find the variance of the leg lengths after his father made the alterations. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 P 9 a Find the mean and standard deviation of the first seven positive even integers. |
b Without using a calculator, write down the mean and standard deviation of the first seven positive odd integers. c Find an expression in terms of n for the variance of the first n positive even integers. What other measure can be found using this expression? 10 Each year Upchester United plays against Upchester City in a local derby match. The number of goals scored in a match by United is denoted by u and the number of goals scored in a match by City is denoted by c. The number of goals scored in the past 15 matches are summarised by 2Σ = c. and c 19 Σ = u( Σ − u( Σ − 9 =, 25 39 1) 1) =, 2 a How many goals have been scored altogether in these 15 matches? 2Σ b Show that u =. 58 c Find, correct to 3 decimal places, the variance of the number of goals scored by the two teams together in these 15 matches. 11 Twenty values of x are summarised by x( Σ − 1) 2 = 132 and x( Σ − 1) = 44. Eighty values of y are summarised by y( Σ + 2 1) = 17 704 and y( Σ + 1) 1184. = a Show that x 64 Σ = 2Σ and that x = 240. b Calculate the value of yΣ and of 2yΣ. 78 c Find the exact variance of the 100 values of x and y combined. 12 The heights, x cm, of 200 boys and the heights, y cm, of 300 girls are summarised by the following totals: x( Σ − 160) 2 = 18 240, x( Σ − 160) 1820 =, y( Σ − 150) 2 = 20 100, y( Σ − 150) =. 2250 a Find the mean height of these 500 children. b By first evaluating 2xΣ and 2yΣ, find the variance of the heights of the 500 children, including appropriate units with your answer. What effect does multiplication of all the values in a dataset have on its variation? And how can we find the variance and standard deviation of the original values from the coded data? Consider the total cost of hiring a taxi for which a customer pays a fixed charge of $3 plus $4 per kilometre travelled. Using y for the total cost and x for the distance travelled in kilometres, the cost can be calculated from the equation y |
are shown in the following table. +. Some example values x4 = 3 1+ → 1+ → Distance x( km) Cost y($ ) 1 7 2 11 3 15 → 4+ → 4+ Copyright Material - Review Only - Not for Redistribution FAST FORWARD You will study the variance of linear combinations of random variables in the Probability & Statistics 2 Coursebook, Chapter 3. REWIND We saw in Chapter 2, Section 2.2 that the mean of a set of data can be found from a coded total such as −. ) ax b Σ ( Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation When the distance changes or varies by 1+, the cost changes or varies by 4+. Variation in cost is affected by multiplication ( 4)× but not by addition ( 3)+. If we consider the graph of y variation of y. If we increase the x-coordinate of a point on the line by 1, its y-coordinate increases by 4. +, then it is only the gradient of the line that affects the x4 = 3 Multiplying x {1, 2, 3} of spread. = by 4 ‘stretches’ the whole set to {4, 8, 12}, which affects the pattern Adding 3+ to {4, 8, 12} simply translates the whole set to y {7, 11, 15} on the pattern of spread. =, which has no effect Journeys of 1, 2 and 3 |
km, costing $7, $11 and $15 are represented in the following diagram 10 11 12 13 14 15 Distance Cost x y = 4x + 3 In the diagram, we see that the range of y is 4 times the range of x, so the range of x is 1 4 times the range of y. For x {1, 2, 3} the following results:, you should check to confirm 3 {7, 11, 15} and x4 + = = SD( x ) = 1 4 × SD(4 x + and 3) Var( x ) = 1 16 × Var(4 x + 3). 79 KEY POINT 3.6 x For ungrouped data For grouped data ( ax b − n − Σ ) ( ax b − n 2 Σ 2 ) f ( ax b − f Σ − Σ ( ) ax b f − f Σ 2 These formulae can be summarised by writing Var( x ) = 1 2 a × Var( ax b − ) or Var( ax b − ) = 2 a × Var( x ). WORKED EXAMPLE 3.14 The standard deviation of the prices of a selection of brand-name products is $24. Imitations of these products are all sold at 25% of the brand-name price. Find the variance of the prices of the imitations. Answer Var(0.25 ) x 2 0.25 Var( × x ) 0.25 2 2 24 × 36 = = = Denoting the brand-name prices by x and 0.25, we use the the imitation prices by 2= ) Var( fact that to find a × Var x x Var( (0.25 ). ax x ) TIP The units for variance in this case are ‘dollars squared’. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 WORKED EXAMPLE 3.15 |
Given that Σ x(3 − 1) 2 = 9136, Σ x(3 − 1) 53 = and n 10=, find the value of x2Σ. Answer 3 x 1 − = x = = 53 10 1 3 2.1 × 53 10 + 1 We first find x, knowing that the mean of the coded values is 1 less than 3 times the mean of x; that is, mean x (3 – 1) 3 x −. 1 = Var( x ) = × Var(3 x − 1) We form and solve an equation knowing that 2 2.1 − = × 9136 10 2 5.3 − Var( x ) 1 2= a × Var( ax b − ) and Var(3 – 1) x = 9136 10 5.32. − 1 2 3 1 9 2 x Σ 10 2 x Σ 10 − 4.41 98.39 = ∴ Σ x 2 = 1028 EXERCISE 3E 80 1 The range of prices of the newspapers sold at a kiosk is $0.80. After 6p.m. all prices are reduced by 20%. Find the range of the prices after 6p.m. 2 Find the standard deviation of x, given that 2Σ x4 = 14600, Σ x2 = 420 and n. 20= 3 The values of x given in the table on the left have a standard deviation of 0.88. Find the standard deviation of the values of y 13 b 19 c 4 The temperatures, °T Celsius, at seven locations in the Central Kalahari Game Reserve were recorded at 4p.m. one January afternoon. The values of T, correct to 1 decimal place, were: 32.1, 31.7, 31.2, 31.5, 31.9, 32.2 and 32.7. a Evaluate Σ T10( − 30) and Σ 100( T 30)2. − b Use your answers to part a to calculate the standard deviation of T. c By 5p.m. the temperature at each location had dropped by exactly 0.75 C. Find the variance of the ° temperatures at 5p.m. 5 Building plots are offered for sale at $315 per square metre. The seller has to pay a lawyer’s fee of $500 from the |
money received. Salome’s plot is 240 square metres larger than Nadia’s plot. How much more did the seller receive from Salome than from Nadia after paying the lawyer’s fees? 6 Temperatures in degrees Celsius ( C)° +. formula F 1.8C 32 = can be converted to temperatures in degrees Fahrenheit ( F)° using the a The temperatures yesterday had a range of °15 C. Express this range in degrees Fahrenheit. b Temperatures elsewhere were recorded at hourly intervals in degrees Fahrenheit and were found to have mean 54.5 and variance 65.61. Find the mean and standard deviation of these temperatures in degrees Celsius. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation M 7 Ten items were selected from each of four sections at a supermarket. Details of the prices of those items, in dollars, on 1st April and on 1st June are shown in the following table. Bakery Household Tinned food Fruit & veg 1st April 1st June Mean 2.50 5.00 4.00 2.00 SD 0.40 1.20 0.60 0.40 Mean 2.25 4.75 4.10 2.00 SD 0.36 1.25 0.60 0.50 For which section’s items could each of the following statements be true? Briefly explain each of your answers. a The total cost of the items did not change. b The price of each item changed by the same amount. c The proportional change in the price of each item was the same. PS 8 The lengths of 45 ropes used at an outdoor recreational centre can be extended by 30% when stretched. The sum of the squares of their stretched lengths is 0.0507 km2 and their natural lengths, x metres, are summarised by x(. Find the mean natural length of these 45 ropes. 1200 20) 2 Σ − = PS 9 Over a short period of time in 2016, the value of the pound sterling (£) fell by 15.25% against the euro (€). Find the percentage change in the value of the euro against the pound over this same period. Appendix to Section 3.3 In this appendix, we show how the two formulae for variance |
are equivalent. For simplicity, we will assume that n 3=. 81 If we denote our three numbers by x x,1 2 and x3, then x )2 Variance is defined by Var( x ) =, so if we expand the brackets and rearrange, we get 1 (= 3 x 1 + x 2 + 3 − ) − 1 3 1 3 1 3 2 x 1 + 2 x 1 + Var + + xx x 1 2 x 2 − 2 xx xx 3 + x 2 + Note: the term in brackets is equal to x. 2 x Σ 3 − x 2, which is the alternative formula 2 x Σ n 2 − x in the case where n 3=. Try showing that the two formulae for variance are equivalent for the simple case where 4= or larger. Can you generalise this n argument to an arbitrary value of n? 2=, and then challenge yourself by taking on n Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Checklist of learning and understanding ● Commonly used measures of variation are the range, interquartile range and standard deviation. ● A box-and-whisker diagram shows the smallest and largest values, the lower and upper quartiles and the median of a set of data. ● For ungrouped data, the median Q2 is at the ● For grouped data with total frequency = Σn th value. n 1 + 2 f, the quartiles are at the following values. ● Lower quartile Q1 is at or Σ. f ● Middle quartile Q2 is at or n 4 n 2 n3 or ● Upper quartile Q3 is at ● IQR = –3 Q Q 1 ● For ungrouped data: 82 E E E Standard deviation = Variance = ), where x − = x Σ n. ● For grouped data: Standard deviation = Variance = ( x x Σ − f Σ ), where − |
x = xf Σ f Σ. ● For datasets x and y with nx and ny values, respectively: 2 Mean = and Variance = . ● The formulae for ungrouped and grouped coded data can be summarised by: Var( x ) Var( – ) x b = and Var( x ) = 1 2 a × Var ( – ax b or ) Var ( – ax b ) = 2 a × Var ( x ) ● For ungrouped coded data − ) ( x b Σ − n 2 and ( ax b − n − Σ ( ) ax b − n 2 ● For grouped coded data − ) ( x b f Σ − f Σ 2 and ( ax b − f Σ − Σ ( ) ax b f − f Σ 2 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation END-OF-CHAPTER REVIEW EXERCISE 3 1 Three boys and seven girls are asked how much money they have in their pockets. The boys have $2.50 each and the mean amount that the 10 children have is $3.90. a Show that the girls have a total of $31.50. b Given that the seven girls have equal amounts of money, find the standard deviation of the amounts that the 10 children have. [1] [3] P 2 Jean Luc was asked to record the times of 20 athletes in a long distance race. He started his stopwatch when the race began and then went to sit in the shade, where he fell asleep. On waking, he found that x athletes had already completed the race but he was able to record the times taken by all the others. State the possible value(s) of x if Jean Luc |
was able to use his data to calculate: a the variance of the times taken by the 20 athletes b the interquartile range of the times taken by the 20 athletes. [1] [2] P 3 The quiz marks of nine students are written down in ascending order and it is found that the range and interquartile range are equal. Find the greatest possible number of distinct marks that were obtained by the nine students. [2] 4 Two days before a skiing competition, the depths of snow, x metres, at 32 points on the course were measured and it was discovered that the numerical values of xΣ and x2Σ were equal. a Given that the mean depth of snow was 0.885 m, find the standard deviation of x. b Snow fell the day before the competition, increasing the depth over the whole course by 1.5cm. Explain what effect this had on the mean and on the standard deviation of x. [2] [2] 83 5 The following box plots summarise the percentage scores of a class of students in the three Mathematics tests they took this term. 30 40 50 Percentage scores (%) 60 70 80 90 100 first test second test third test a Describe the progress made by the class in Mathematics tests this term. b Which of the tests has produced the least skewed set of scores? c What type of skew do the scores in each of the other two tests have? 6 The following table shows the mean and standard deviation of the lengths of 75 adult puff adders (Bitis arietans), which are found in Africa and on the Arabian peninsula. Frequency Mean (cm) SD (cm) African Arabian 60 15 102.7 78.8 6.8 4.2 a Find the mean length of the 75 puff adders. b The lengths of individual African puff adders are denoted by x f and the lengths of individual Arabian puff adders by xb. By first finding x f 75 puff adders. 2Σ and xb 2Σ, calculate the standard deviation of the lengths of all [2] [1] [2] [3] [5] Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 M 7 The scores obtained by 11 people throwing |
three darts each at a dartboard are 54, 46, 43, 52, 180, 50, 41, 56, 52, 49 and 54. a Find the range, the interquartile range and the standard deviation of these scores. b Which measure in part a best summarises the variation of the scores? Explain why you have chosen this particular measure. [4] [2] 8 The heights, x cm, of a group of 28 people were measured. The mean height was found to be 172.6cm and the standard deviation was found to be 4.58cm. A person whose height was 161.8cm left the group. i Find the mean height of the remaining group of 27 people. ii Find x2Σ for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people. [2] [4] 9 120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table. Cambridge International AS & A Level Mathematics 9709 Paper 63 Q4 June 2014 Time (seconds) 1–25 26–35 36– 45 46–55 56–90 Number of people 4 24 38 34 20 84 Calculate estimates of the mean and standard deviation of the reading times. [5] Cambridge International AS & A Level Mathematics 9709 Paper 62 Q2 June 2015 10 The weights, in kilograms, of the 15 basketball players in each of two squads, A and B, are shown below. Squad A Squad B 97 75 98 104 84 100 109 115 99 122 82 116 96 84 107 91 79 94 101 96 77 111 108 83 84 86 115 82 113 95 i Represent the data by drawing a back-to-back stem-and-leaf diagram with squad A on the left-hand side [4] of the diagram and squad B on the right-hand side. ii Find the interquartile range of the weights of the players in squad A. [2] iii A new player joins squad B. The mean weight of the 16 players in squad B is now 93.9kg. Find the weight [3] of the new player. Cambridge International AS & A Level Mathematics 9709 Paper 62 Q5 November 2015 [Adapted] 11 The heights, x cm, of a group of 82 children are summarised as follows. x( Σ − 130) = − 287., standard deviation of x 6.9= i Find |
the mean height. ii Find x( Σ − 130)2. [2] [2] Cambridge International AS & A Level Mathematics 9709 Paper 63 Q2 June 2010 12 A sample of 36 data values, x, gave x( Σ − 45) = − 148 and x( Σ − 45) 2 =. 3089 i Find the mean and standard deviation of the 36 values. ii One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values. [3] [4] Cambridge International AS & A Level Mathematics 9709 Paper 62 Q3 June 2011 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 3: Measures of variation 13 The ages, x years, of 150 cars are summarised by x 645 Σ = 2Σ and x = 8287.5. Find ( Σ − x x the mean of x. )2, where x denotes [4] Cambridge International AS & A Level Mathematics 9709 Paper 62 Q1 June 2012 M 14 A set of data values is 152, 164, 177, 191, 207, 250 and 258. Compare the proportional change in the standard deviation with the proportional change in the interquartile range when the value 250 in the data set is increased by 40%. [5] PS 15 A shop has in its stock 80 rectangular celebrity posters. All of these posters have a width to height ratio of 1: 2, and their mean perimeter is 231.8cm. Given that the sum of the squares of the widths is 200120 cm2, find the standard deviation of the widths of the [4] posters. P PS 16 At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were 180, 211, 230, 199, 214 and 166. a Show that the mean of these guesses is 200, and use SD = )2 ( x x Σ − n to calculate the standard deviation. [4] b The jar actually contained 202 sweets. Without further calculation, write down the mean and the standard deviation of the errors made by these six visitors. Explain why no further calculations are required to do this. [4] 17 The number of women in senior management positions at a number of companies was investigated |
. The number of women at each of the 25 service companies and at each of the 16 industrial companies are denoted by Sw and 2 5) − Σ Iw, respectively. The findings are summarised by the totals: 2, w 3 ) = Σ I and w 3 ) ( IΣ, w 5 ( SΣ ) 15 = 4 = −. w( S 28 12 − − − = ( 85 [3] [5] [3] [3] [7] a Show that there are, on average, more than twice as many women in senior management positions at the service companies than at the industrial companies. 2 b Show that w S Σ ≠ Σ w( S ) 2 2 and that w I Σ ≠ Σ w( I ) 2. c Find the standard deviation of the number of women in senior management positions at all of these service and industrial companies together. 18 The ages, a years, of the five members of the boy-band AlphaArise are such that a( Σ − 21) 2 = 11.46 and a( Σ − 21) = −. 6 The ages, b years, of the seven members of the boy-band BetaBeat are such that b( b( Σ − =. 0 18) Σ − 2 18) = 10.12 and a Show that the difference between the mean ages of the boys in the two bands is 1.8 years. b Find the variance of the ages of the 12 members of these two bands. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Cambridge International AS & A Level Mathematics: Probability & Statistics 1 CROSS-TOPIC REVIEW EXERCISE 1 1 Two players, A and B, both played seven matches to reach the final of a tennis tournament. The number of games that each of them won in these matches are given in the following back-to-back stem-and-leaf diagram. Player Player B 8 9 1 1 3 4 6 Key: 5 2 6 represents 25 games for A and 26 games for B a How many fewer games did player B win than player A? b Find the median number of games won by each player. c |
In a single stem-and-leaf diagram, show the number of games won by these two players in all of the 14 matches they played to reach the final. 2 A total of 112 candidates took a multiple-choice test that had 40 questions. The numbers of correct answers given by the candidates are shown in the following table. No. correct answers No. candidates 0–9 18 10–15 16–25 26–30 31–39 24 27 23 19 40 1 a State which class contains the lower quartile and which class contains the upper quartile. Hence, find the least possible value of the interquartile range. b Copy and complete the following table, which shows the numbers of incorrect answers given by 86 the candidates in the test. No. incorrect answers No. candidates 0 1 1–9 c Calculate an estimate of the mean number of incorrectly answered questions. 3 At a factory, 50-metre lengths of cotton thread are wound onto bobbins. Due to fraying, it is common for a length, l cm, of cotton to be removed after it has been wound onto a bobbin. The following table summarises the lengths of cotton thread removed from 200 bobbins. Length removed ( cm) l 0 < l 2.5< 2.5 < l 5.0< No. bobbins 137 49 5 < l 10< 14 a Calculate an estimate of the mean length of cotton removed. b Use your answer to part a to calculate, in metres, an estimate of the standard deviation of the length of cotton remaining on the 200 bobbins. 4 People applying to a Computing college are given an aptitude test. Those who are accepted take a progress test 3 months after the course has begun. The following table gives the aptitude test scores, x, and the progress test scores, y, for a random sample of eight students, A to H. x y A 61 53 B 80 77 C 74 61 D 60 70 E 83 81 F 92 54 G 71 63 H 67 85 a Find the interquartile range of these aptitude test scores. b Use the summary totals Σ =x 588, Σ, Σ =y to calculate the variance of the aptitude and progress test scores when they are considered together. 544 and Σ 44 080 38030 2 =x 2 =y [1] [2] [3] [3] [3] [3] [3] [4] [1] [3] Copyright Material |
- Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cross-topic review exercise 1 c The mean progress score for all the students at the college is 70 and the variance is 112. Any student who scores less than 1.5 standard deviations below the mean is sent a letter advising that improvement is needed. Which of the students A to H should the letter be sent to? [1] 5 The growth of 200 tomato plants, half of which were treated with a growth hormone, was monitored over a 5-day period and is summarised in the following graphs 100 80 60 40 20 0 Treated Untreated 1 2 3 4 5 6 7 8 9 10 Growth (cm) 87 Use the graphs to describe two advantages of treating these tomato plants with the growth hormone. [2] 6 A survey of a random sample of 23 people recorded the number of unwanted emails they received in a particular week. The results are given below. 9 18 13 18 21 17 22 27 8 11 26 26 32 17 31 20 36 15 13 25 35 29 14 a Represent the data in a stem-and-leaf diagram. b Draw, on graph paper, a box-and-whisker diagram to represent the data. 7 The volumes of water, x 106× 2.82, 2.50, 2.75, 3.14, 3.66 and 3.07. litres, needed to fill six Olympic-sized pools are a Find the value of Σ −x( 2) and of Σ −x( 2)2. b Use your answers to part a to find the mean and the standard deviation of the volumes of water, giving both answers correct to the nearest litre. 8 The speeds of 72 coaches at a certain point on their journeys between two cities were recorded. The results are given in the following table. Speed (km/h) Cumulative frequency < 50 0 < 54 9 < 70 41 < 75 54 < 85 72 a State the number of coaches whose speeds were between 54 and 70 km/h. b A student has illustrated the data in a cumulative frequency polygon. Find the two speeds between which the polygon has the greatest gradient. c Calculate an estimate of the lower boundary of the speeds of the fastest 25 coaches. [3] [4] [2] [5] |
[1] [1] [3] Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Cambridge International AS & A Level Mathematics: Probability & Statistics 1 9 The following table shows the masses, m grams, of 100 unsealed bags of plain potato crisps. Mass (m grams) No. bags ( )f 34.6 < m < 35.4 35.4 < m < 36.2 36.2 < m < 37.2 20 30 50 a Show that the heights of the columns in a histogram illustrating these data must be in the ratio 2 : 3 : 4. b Calculate estimates of the mean and of the standard deviation of the masses. c Before each bag is sealed, 0.05 grams of salt is added. Find the variance of the masses of the sealed bags of salted potato crisps. 10 The masses, in carats, of a sample of 200 pearls are summarised in the following cumulative frequency graph. One carat is equivalent to 200 milligrams. 88 ) 200 160 120 80 40 0 20 40 60 80 100 120 Mass (carats) a Use the graph to estimate, in carats: i the median mass of the pearls ii the interquartile range of the masses. b To qualify as a ‘paragon’, a pearl must be flawless and weigh at least 20 grams. Use the graph to estimate the largest possible number of paragons in the sample. 11 The amounts spent, S dollars, by six customers at a hairdressing salon yesterday were as follows. 12.50, 15.75, 41.30, 34.20, 10.80, 40.85. Each of the customers paid with a $50 note and each received the correct change, which is denoted by C$. a Find, in dollars, the value of S C+ and of S C−. b Explain why the standard deviation of S and the standard deviation of C are identical. Copyright Material - Review Only - Not for Redistribution [2] [4] [1] [1] [2] [2] [3] [3] Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review |
Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cross-topic review exercise 1 12 The numbers of goals, x, scored by a team in each of its previous 25 games are summarised by the totals x( Σ − 1) 2 = 30 and Σ − x( 1) 12. = a Find the mean number of goals that the team scored per game. b Find the value of Σx2. c Find the value of a and of b in the following table, which shows the frequencies of the numbers of goals scored by the team. No. goals No. games ( )> 0 13 The lengths of some insects of the same type from two countries, X and Y, were measured. [2] [3] [2] The stem-and-leaf diagram shows the results. Country X Country Y (10) (18) (16) (16) (11 80 81 82 83 84 85 86 13) (15) (17) (15) (12) (11) Key: 5 81 3 means an insect from country X has length 0.815 cm and an insect from country Y has length 0.813 cm. i Find the median and interquartile range of the lengths of the insects from country X. [2] 89 ii The interquartile range of the lengths of the insects from country Y is 0.028cm. Find the values of q and r. [2] iii Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper. [4] iv Compare the lengths of the insects from the two countries. [2] Cambridge International AS & A Level Mathematics 9709 Paper 63 Q6 June 2010 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 90 Chapter 4 Probability In this chapter you will learn how to: elementary events ■ evaluate probabilities by means of enumeration of equiprobable (i.e. equally likely) ■ use addition and multiplication of probabilities appropriately ■ use the terms mutually exclusive and independent events ■ determine whether two events are independent ■ calculate and use conditional probabilities. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - |
Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability PREREQUISITE KNOWLEDGE Where it comes from What you should be able to do Check your skills IGCSE / O Level Mathematics Calculate the probability of a single event as either a fraction, decimal or percentage. Understand and use the probability scale from 0 to 1. Understand relative frequency as an estimate of probability. Calculate the probability of simple combined events, using possibility diagrams and tree diagrams where appropriate. Use language, notation and Venn diagrams to describe sets and represent relationships between sets. 1 How many 6s are expected when an ordinary fair die is rolled 180 times? 2 Find the probability of obtaining a total of 4 when the scores on two ordinary fair dice are added together. 3 It is given that {2, 4, 5} =A and {1, 2, 5} ′ =A ′ =B a Venn diagram, or otherwise, find n( n(, {3, 4}. Using ) and A B ∪ ′ ′ ∩A B ). 91 If we do this, how likely is that? Probability measures the likelihood of an event occurring on a scale from 0 (i.e. impossible) to 1 (i.e. certain). We write this as P(name of event), and its value can be expressed as a fraction, decimal or percentage. The greater the probability, the more likely the event is to occur. Although we do not often calculate probabilities in our daily lives, we frequently assess and compare them, and this affects our behaviour. Do we have a better chance of performing well in an exam after a good night’s sleep or after revising late into the night? Should you visit a doctor or is your sore throat likely to heal by itself soon? Insurance is based on risk, which in turn is based on the probability of certain events occurring. Government spending is largely determined by the probable benefits it will bring to society. 4.1 Experiments, events and outcomes The result of an experiment is called an outcome or elementary event, and a combination of these is known simply as an event. Rolling an ordinary fair die is an experiment that has six possible outcomes: 1, 2, 3, 4, 5 or 6. Obtaining an odd number with the die is an event that has three favourable outcomes |
: 1, 3 or 5. Random selection and equiprobable events The purpose of selecting objects at random is to ensure that each has the same chance of being selected. This method of selection is called fair or unbiased, and the selection of any particular object is said to be equally likely or equiprobable. KEY POINT 4.1 When one object is randomly selected from n objects, P(selecting any particular object) 1 =. n Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 The probability that an event occurs is equal to the proportion of equally likely outcomes that are favourable to the event. KEY POINT 4.2 P(event) = Number of favourable equally likely outcomes Total number of equally likely outcomes Consider randomly selecting 1 student from a group of 19, where 11 are boys and eight are girls. There are 19 possible outcomes: 11 are favourable to the event selecting a boy and eight are favourable to the event selecting a girl, as shown in the following table. Event/outcome Probability Description TIP Selecting any particular boy Selecting any particular girl Selecting any particular student 92 Selecting a boy Selecting a girl 1 19 1 19 1 19 11 19 8 19 These three outcomes are equally likely. 11 of the 19 equally likely outcomes are favourable to this event. 8 of the 19 equally likely outcomes are favourable to this event. Exhaustive events A set of events that contains all the possible outcomes of an experiment is said to be exhaustive. In the special case of event A and its complement, not A, the sum of their probabilities is 1 because one of them is certain to occur. Recall that the notation used for the complement of set A is A′. The word particular specifies one object. It does not matter whether that object is a boy, a girl or a student; each has a 1 19 chance of being selected. KEY POINT 4.3 A ) 1 = P( A + ) P(not or A ) P( + A ) 1 ′ = Examples of complementary exhaustive events are shown in the following table. P( Experiment Exhaustive events A A′ Toss a fair coin heads tails Roll a fair die less than 2 Play a game of chess win 2 or more not |
win Probabilities (win) P(not win) 1 = + Trials and expectation Each repeat of an experiment is called a trial. The proportion of trials in which an event occurs is its relative frequency, and we can use this as an estimate of the probability that the event occurs. If we know the probability of an event occurring, we can estimate the number of times it is likely to occur in a series of trials. This is a statement of our expectation. Copyright Material - Review Only - Not for Redistribution KEY POINT 4.4 In n trials, event A is expected to occur n times. ) AP( × Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability WORKED EXAMPLE 4.1 The probability of rain on any particular day in a mountain village is 0.2. On how many days is rain not expected in a year of 365 days? Answer 365 n = and P(does not rain) 1 – 0.2 = = 0.8 365 × 0.8 = 292 days We multiply the probability of the event by the number of days in a year. M EXPLORE 4.1 We can see how closely expectation matches with what happens in practice by conducting simple experiments using a fair coin (or an ordinary fair die). Toss the coin 10 times and note as a decimal the proportion of heads obtained. Repeat this and note the proportion of heads obtained in 20 trials. Continue doing this so that you have a series of decimals for the proportion of heads obtained in 10, 20, 30, 40, 50, … trials. Represent these proportions on a graph by plotting them against the total number of trials conducted. How do your results compare with the expected proportion of heads? For trials with a die, draw a graph to represent the proportions of odd numbers obtained. EXERCISE 4A 1 A teacher randomly selects one student from a group of 12 boys and 24 girls. Find the probability that the teacher selects: a a particular boy b a girl. 2 United’s manager estimates that the team has a 65% chance of winning any particular game and an 85% chance of not drawing any particular game. a What are the manager’s estimates most likely to be based on? b If the team plays 40 games this season, find the manager’s expectation of the |
number of games the team will lose. c If the team loses one game more than the manager expects this season, explain why this does not necessarily mean that they performed below expectation. 3 Katya randomly picks one of the 10 cards shown If she repeats this 40 times, how many times is Katya expected to pick a card that is not blue and does not have a letter B on it? Copyright Material - Review Only - Not for Redistribution 93 Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 4 A numbered wheel is divided into eight sectors of equal size, as shown. The wheel is spun until it stops with the arrow pointing at one of the numbers. Axel decides to spin the wheel 400 times. a Find the number of times the arrow is not expected to point at a 4. b How many more times must Axel spin the wheel so that the expected number of times that the arrow points at a 4 is at least 160 bag contains black and white counters, and the probability of selecting a black counter is 1 6. a What is the smallest possible number of white counters in the bag? b Without replacement, three counters are taken from the bag and they are all black. What is the smallest possible number of white counters in the bag? 6 When a coin is randomly selected from a savings box, each coin has a 98% chance of not being selected. How many coins are in the savings box? PS 7 A set of data values is 8, 13, 17, 18, 24, 32, 34 and 38. Find the probability that a randomly selected value is more than one standard deviation from the mean. PS 8 One student is randomly selected from a school that has 837 boys. The probability that a girl is selected is 4 7 particular boy is selected.. Find the probability that a 94 REWIND We studied the mean in Chapter 2, Section 2.2 and standard deviation in Chapter 3, Section 3.3. 4.2 Mutually exclusive events and the addition law To find the probability that event A or event B occurs, we can simply add the probabilities of the two events together, but only if A and B are mutually exclusive. Mutually exclusive events have no common favourable outcomes, which means that it is not possible for both events to occur, so A P( and |
) B 0. = For example, when we roll an ordinary die, the events ‘even number {2, 4, 6} of 5 {1, 5} = It is not possible to roll a number that is even and a factor of 5. We say that the intersection of these two sets is empty. Therefore: ’ are mutually exclusive because they have no common favourable outcomes. ’ and ‘factor = P(evenor factor of 5) P(even) P(factor of 5) = + Events are not mutually exclusive if they have at least one common favourable outcome, which means that it is possible for both events to occur, so P( and ) A B ≠. 0 For example, when we roll an ordinary die, the events ‘odd number {1, 3, 5} ‘factor of 5 {1, 5} outcomes. It is possible to roll a number that is odd and a factor of 5. We say that the intersection of these two sets is not empty. Therefore: ’ are not mutually exclusive because they do have common favourable ’ and = = P(odd or factor of 5) P(odd) P(factor of 5) ≠ + Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability KEY POINT 4.5 The addition law for mutually exclusive events is A B P( or ) P( = A ) P( + B ). This can be extended for any number of mutually exclusive events: P(A or B or C or …) B P( A C ) P( + ) P( + + … = ) Venn diagrams Venn diagrams are useful tools for solving problems in probability. We can use them to show favourable outcomes or the number of favourable outcomes or the probabilities of particular events. The number of outcomes favourable to event A is denoted by An( ). The set of outcomes that are not favourable to event A is the complement of A, denoted by ′A. The following Venn diagrams illustrate various sets and their complements. TIP The universal set ℰ represents the complete set of outcomes and is called the possibility space TIP A Aʹ not A A ∪ B A or B (A ∪ B)� |
� neither A nor B A ∩ B A and B (A ∩ B)ʹ not both A and B KEY POINT 4.6 ‘A or B’ means event A occurs or event B occurs or both occur. ∪A B means ‘A or B’ and ∩A B means ‘A and B’. 95 Using set notation, the addition law for two mutually exclusive events is A B P( ∪ = ) P( A ) P( + B. ) A and B are mutually exclusive when A B empty set). P( ∩ = ; that is, when A B∩ = ∅ (∅ means the ) 0 For non-mutually exclusive events, P( )∪A B can be found by enumerating (counting) the favourable equally likely outcomes, taking care not to count any of them twice. We show how this can be done in part b of the following example. WORKED EXAMPLE 4.2 One digit is randomly selected from 1, 2, 3, 4, 5, 6, 7, 8 and 9. Three possible events are: A: a multiple of 3 is selected. B: a factor of 8 is selected. C: a prime number is selected. a Show that the only pair of mutually exclusive events from A, B and C is A and B, and find P( )∪A B. b Find: i P( )∪A C ii P( )∪B C. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Answer = ℰ {1, 2, 3, 4, 5, 6, 7, 8, 9} 3 =. 9 A {3, 6, 9} ), so AP( = B {1, 2, 4, 8} =, so BP( ) C {2, 3, 5, 7} =, so CP∩ = ∅, so A and B are mutually exclusive. A C∩ ≠ ∅, so A and C are not mutually exclusive. B C∩ ≠ ∅, so B and C are not mutually exclusive. P( 96 A B � |
� = ) P( or A B ) B ) ) P( 3 9 7 9 By listing and counting the favourable outcomes, we can find the probability for each event. Outcomes favourable to pairs of events are shown in the three Venn diagrams opposite. Two events are mutually exclusive when they have no common favourable outcomes; that is, when their intersection is an empty set. We can use the addition law because A and B are mutually exclusive events. b Both parts of this question can be answered using the lists of elements or the previous Venn diagrams. i n( A C ∪ ) n( = A ) n( + C ) n( − A C ∩ ) P( A C ∪ ) P( = A ) P( + C ) P or 2 3 ii n( B ∪ C ) n( = B ) n( + C ) n( − B ∩ C ) P( B ∪ C ) P( = B ) P( + C ) P In the first part of b above, we subtracted ( common elements in ∩A C n( ) have been counted in An( ∩A C ) because the ) and in Cn( ). We follow the same steps when working directly with probabilities. This also applies to events that are mutually exclusive, where the number of common elements is equal to zero. So in general, for any two events A and B: n( A B ∪ = ) n( A ) n( + B ) n( − A B ∩ ) and Set A contains 3 of the 9 elements. Set C contains 4 of the 9 elements. Set A and set C have 1 of the 9 elements in common. Set B contains 4 of the 9 elements. Set C contains 4 of the 9 elements. Set B and set C have 1 of the 9 elements in common. P( A B ∪ = ) P( A ) P( + B ) P( − A B ∩ ). Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy WORKED EXAMPLE 4.3 In a survey, 50% of the participants own a desktop ( 15% own both. )D, 60% own a laptop ( )L and What percentage of the participants owns neither a desktop nor a laptop? |
Answer 1 0.5 D L p 0.15 q 0.6 x The Venn diagram shows the given information, where,p q and x represent, respectively, the percentage that own a desktop only, a laptop only and neither of these. p = 0.5 – 0.15 = 0.35 q = 0.6 – 0.15 = 0.45 x 1 – (0.35 = + 0.15 + 0.45) = 0.05 or 5% 5%∴ of the participants own neither a desktop nor a laptop. Chapter 4: Probability TIP The symbol ∴ means ‘therefore’. WORKED EXAMPLE 4.4 97 Forty children were each asked which fruits they like from apples ( bananas ( )B and cherries ( )A, )C. The following Venn diagram shows the number of children that like each type of fruit. Find the probability that a randomly selected child likes apples or bananas. Answer 40 17 A 6 5 3 7 1 2 C 14 2 B 24 8 P( A B ∪ = ) P( A B ) – P( A B ∩ ) =An( ) 17, =Bn( ) 8 and n( A B ) ∩ = 4. ) P( + 8 40 − + = = 17 40 21 40 4 40 n( A B ∪ ≠ ) 17 + 8 because A B )∩ ≠ ∅. ( There are 17 like apples or bananas. 8 – 4 = + 21 children who Alternatively, we can add up the numbers in the A and B circles 21. 2 KEY POINT 4.7 For any two events, A and B, P( or A B ) P( = A B ∪ = ) P( A ) P( + B ) – P( A B ∩ ). Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 EXERCISE 4B 1 Find the probability that the number rolled with an ordinary fair die is: a a prime number or a 4 b a square number or a multiple of 3 c more than 3 or a factor of 8. 2 A group of 40 students took a test in Economics. The following Venn |
)B took the test and that seven students diagram shows that 19 boys ( failed the test ( )F. 40 B F a Describe the 21 students who are members of the set B′. 16 3 4 b Find the probability that a randomly selected student is a boy or someone who failed the test. 17 3 The following table gives information about all the animals on a farm. Male Female Goats Sheep 5 3 25 22 a Find the probability that a randomly selected animal is: i male or a goat ii a sheep or female. b Find a different way of describing each of the two types of animal in part a. 98 4 Two ordinary fair dice are rolled and three events are: X : the sum of the two numbers rolled is 6. Y : the difference between the two numbers rolled is zero. Z: both of the numbers rolled are even. a List the outcomes that are favourable to: i X and Y ii X and Z iii Y and Z. b What do your answers to part a tell you about the events,X Y and Z? 5 The letters A, B, B, B, C, D, D and E are written onto eight cards and placed in a bag. Find the probability that the letter on a randomly selected card is: a a vowel or in the word DOMAIN b a consonant or in the word DOUBLE. 6 In a group of 25 boys, nine are members of the chess club ( are members of the debating club ( )D and 10 are members of neither of these clubs. This information is shown in the Venn diagram. )C, eight 25 9 C D a b c 8 a Find the values of,a b and c. b Find the probability that a randomly selected boy is: 10 i a member of the chess club or the debating club ii a member of exactly one of these clubs. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability 7 Forty girls were asked to name the capital of Cuba and of Hungary; 19 knew the capital of Cuba, 20 knew the capital of Hungary and seven knew both. a Draw a Venn diagram showing the number of girls who knew each of these capitals. b Find the probability that a randomly selected girl knew: i the capital of Cuba but not of Hungary ii |
just one of these capitals. 8 In a survey on pet ownership, 36% of the participants own a cat, 20% own a hamster but not a cat, and 8% own a hamster and a cat. What percentage of the participants owns neither a hamster nor a cat? 9 A garage repaired 132 vehicles last month. The number of vehicles that required electrical ( diagram opposite. )E, mechanical ( )M and bodywork ( )B repairs are given in the Find the probability that a randomly selected vehicle required: a mechanical or bodywork repairs b no bodywork repairs c exactly two types of repair. E 12 11 13 5 26 M 47 18 B P 33 S 17 32 7 11 M 99 10 The 100 students at a technical college must study at least one subject from Pure Mathematics ( studying these subjects are given in the diagram opposite. )S and Mechanics ( )P, Statistics ( )M. The numbers a Who does the number 17 in the diagram refer to? b Find the probability that a randomly selected student studies: i Pure Mathematics or Mechanics ii exactly two of these subjects. c List the three subjects in ascending order of popularity. 11 Events X and Y are such that P( X ) = 0.5, P( Y ) = 0.6 and P( X Y∩ = ) 0.2. a State, giving a reason, whether events X and Y are mutually exclusive. b Using a Venn diagram, or otherwise, find P( X Y∪. ) c Find the probability that X or Y, but not both, occurs. 12,A B and C are events where P( ) A = 0.3, P( B ) = 0.4, P( C ) = 0.3, P( A B ) ∩ = 0.12, P( A C ) ∩ = 0 and P( B C ) ∩ = 0.1. a State which pair of events from,A B and C is mutually exclusive. b Using a Venn diagram, or otherwise, find P[( ∪ ∪ ′, ) ] which is the probability that neither A nor B nor C occurs. A B C 13 The diagram opposite shows a 30 cm square board with two rectangular cards attached. The 15 cm by 20 cm card covers one-quarter of the 8cm by 12 cm card. A dart is randomly thrown at the board, so that it sticks within its perimeter. Use areas to calculate the probability that the dart pierces |
: 30 a both cards b at least one of the cards c exactly one of the cards. Copyright Material - Review Only - Not for Redistribution 8 12 15 20 30 Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 ) 14 Given that A P( = 0.4, P( B ) = 0.7 and that A B ) ∪ = P( 0.8, find: a ∪ ′BAP( ) b P( ′ ∩A B ) PS 15 Each of 27 tourists was asked which of the countries Angola ( )A, Burundi ( )B and Cameroon ( )C they had visited. Of the group, 15 had visited Angola; 8 had visited Burundi; 12 had visited Cameroon; 2 had visited all three countries; and 21 had visited only one. Of those who had visited Angola, 4 had visited only one other country. Of those who had not visited Angola, 5 had visited Burundi only. All of the tourists had visited at least one of these countries. a Draw a fully labelled Venn diagram to illustrate this information. b Find the number of tourists in set B′ and describe them. c Describe the tourists in set ( ) A B C ∪ ∩ ′ and state how many there are. d Find the probability that a randomly selected tourist from this group had visited at least two of these three countries. 4.3 Independent events and the multiplication law Two events are said to be independent if either can occur without being affected by the occurrence of the other. Examples of this are making selections with replacement and performing separate actions, such as rolling two dice. KEY POINT 4.8 100 The multiplication law for independent events is A P( and ) P( B = A B ∩ ) P( = A ) P( × B. ) This can be extended for any number of independent events: P(A and B and C and …) = P( A B C ∩ ∩ ∩ =...) P( A ) P( × B ) P( × C ) ×... Consider the following bag, which contains two blue balls ( and five white balls ( )W. )B We will select one ball at random, replace it and then select another ball. For |
the first selection: BP( ) = and WP( 2 7 ) 5 =. 7 For the second selection: BP( 5 =. 7 The tree diagram below shows how we can use the multiplication law to find probabilities. = and WP( 2 7 ) ) 1st 1st B W 2 7 5 7 2nd Events and probabilities....... P(BB) B W....... P(BW)....... P(WB) B W....... P(WW 49 10 49 10 49 25 49 2 7 5 7 2 7 5 7 Copyright Material - Review Only - Not for Redistribution TIP The first and second selections are made from the same seven balls, so probabilities are identical and independent. TIP We can denote the event ‘2 blue balls are selected’ by BB; B and B; &B B or,B B. TIP,, BB BW WB Events and WW are exhaustive, so their probabilities sum to 1. Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Multiplication of independent events is performed from left to right along the branches. Addition of mutually exclusive events is performed vertically. As examples: P(different colours) P( = or BW WB ) P( = BW ) P( + WB ) = P(same colours) P( = or BB WW ) P( = BB ) P( + WW ) = 2 × 7 5 7 + 5 × 7 2 7 2 × 7 2 7 + 5 × 7 5 7 = = 10 49 4 49 + + 10 49 25 49 = = 20 49 29 49 As an alternative to using a tree diagram, we can use a possibility diagram (or outcome space), as shown below. The diagram shows the 7 four mutually exclusive combined events 7 × = BB BW WB and WW., equally likely outcomes and the 49, BW BW WW WW WW WW WW BW BW WW WW WW WW WW BW BW WW WW WW WW WW BW BW WW WW WW WW WW BW BW WW WW WW WW WW BB BB WB WB WB WB WB BB BB WB WB WB WB WB B B WWWW W 1st selection To |
find probabilities for combined events, we count how many of the 49 outcomes are favourable. For example: P(at least 1 blue) P( = BB ) P( + BW ) P( + WB ) = 4 49 + 10 49 + 10 49 = 24 29. Alternatively, P(at least 1 blue) 1 P( = − WW ) 1 = − 25 49 = 24 29. Chapter 4: Probability TIP If we just used a 2 by 2 diagram, with B and W as the outcomes of each selection, we could not just count cells to find probabilities, because the events in those four cells would not be equally likely. TIP Probabilities are equal to the relative frequencies of the favourable outcomes. 101 WORKED EXAMPLE 4.5 Find the probability that the sum of the scores on three rolls of an ordinary fair die is less than 5. Answer P(sum<5) P(3) P(4) = + The lowest possible sum is 3. P(sum 3) = = P(sum 4) = = 1 216 3 216 P(sum 5) < = 1 216 + 3 216 = 1 54 Each roll has six equiprobable outcomes, so there are 6 216 possible outcomes, each with a probability 6 = 6 × × 1 216. of There is one way to obtain a sum of 3: (1, 1, 1). There are three ways to obtain a sum of 4: (1, 1, 2), (1, 2, 1), (2, 1, 1). Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 WORKED EXAMPLE 4.6 Abha passes through three independent sets of traffic lights when she drives to work. The probability that she has to stop at any particular set of lights is 0.2. Find the probability that Abha: a first has to stop at the second set of lights b has to stop at exactly one set of lights c has to stop at any set of lights. Answer a P(XS) = 0.8 × 0.2 = 0.16 b P(has to stop at exactly 1 set of lights) P(SXX) P(XSX) P(XXS) |
+ + = (0.2 × 0.8 × 0.8) 3 = × = 0.384 c P(has to stop) 1 – P (does not have to stop) = = 1 – P(XXX) = − 1 0.8 3 102 0.488 = Alternatively, P(has to stop) = EXERCISE 4C P(S) P(XS) P(XXS). + + We use S and X to represent ‘stopping’ and ‘not stopping’. For each set of lights, P(S) and P(X). 0.8= 0.2= The three favourable outcomes, SXX, XSX and XXS, are equally likely. The events ‘has to stop’ and ‘does not have to stop’ are complementary. 1 Using a tree diagram, find the probability that exactly one head is obtained when two fair coins are tossed. 2 Two ordinary fair dice are rolled. Using a possibility diagram, find the probability of obtaining: a two 6s b two even numbers c two numbers whose product is 6. 3 It is known that 8% of all new FunX cars develop a mechanical fault within a year and that 15% independently develop an electrical fault within a year. Find the probability that within a year a new FunX car develops: a both types of fault b neither type of fault. 4 A certain horse has a 70% chance of winning any particular race. Find the probability that it wins exactly one of its next two races. 5 The probabilities that a team wins, draws or loses any particular game are 0.6, 0.1 and 0.3, respectively. a Find the probability that the team wins at least one of its next two games. b If 2 points are awarded for a win, 1 point for a draw and 0 points for a loss, find the probability that the team scores a total of more than 1 point in its next two games. 6 On any particular day, there is a 30% chance of snow in Slushly. Find the probability that it snows there on: a none of the next 3 days b exactly one of the next 3 days. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4 |
: Probability M 7 Fatima will enter three sporting events at the weekend. Her chances of winning each of them are shown in the following table. Event Shot put Javelin Discus Chance of winning 85% 40% 64% a Assuming that the three events are independent, find the probability that Fatima wins: i the shot put and discus ii the shot put and discus only iii exactly two of these events. b What does ‘the three events are independent’ mean here? Give a reason why this may not be true in real life. 8 A fair six-sided spinner, P, has edges marked 0, 1, 2, 2, 3 and 4. A fair four-sided spinner, Q, has edges marked 0, –1, –1 and –2. Each spinner is spun once and the numbers on which they come to rest are added together to give the score, S. Find: a SP( 2)= b SP( 2 = 1) M 9 Letters and packages can take up to 2 days to be delivered by Speedipost couriers. The following table shows the percentage of items delivered at certain times after sending. Same day After 1 day After 2 days Letter Package 40% 15% 50% 55% 10% 30% a Is there any truth in the statement ‘If you post 10 letters on Monday then only nine of them will be delivered before Wednesday’? Give a reason for your answer. b Find the probability that when three letters are posted on Monday, none of them are delivered on Tuesday. c Find the probability that when a letter and a package are posted together, the letter arrives at least 1 day before the package. 103 10 The following histogram represents the results of a national survey on bus departure delay times. Two buses are selected at random. Calculate an estimate of the probability that: a both departures were delayed by less than 4 minutes b at least one of the buses departed more than 7 minutes late % ( 20 15 10 10 Delay time (min) 11 Praveen wants to speak on the telephone to his friend. When his friend’s phone rings, he answers it with constant probability 0.6. If Praveen’s friend doesn’t answer his phone, Praveen will call later, but he will only try four times altogether. Find the probability that Praveen speaks with his friend: a after making fewer than three calls b on the telephone on this occasion. Copyright Material - Review Only |
- Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 12 Each morning, Ruma randomly selects and buys one of the four newspapers available at her local shop. Find the probability that she buys: a b the same newspaper on two consecutive mornings three different newspapers on three consecutive mornings. 13 A coin is biased such that the probability that three tosses all result in heads is obtaining no heads with three tosses of the coin. 125 512. Find the probability of 14 In a group of five men and four women, there are three pairs of male and female business partners and three teachers, where no teacher is in a business partnership. One man and one woman are selected at random. Find the probability that they are: a both teachers b in a business partnership with each other c each in a business partnership but not with each other. PS 15 A biased die in the shape of a pyramid has five faces marked 1, 2, 3, 4 and 5. The possible scores are 1, 2, 3, 4 and 5 and x P( ) = k x − 25, where k is a constant. a Find, in terms of k, the probability of scoring: i 5 ii less than 3. b The die is rolled three times and the scores are added together. Evaluate k and find the probability that the 104 sum of the three scores is less than 5. PS 16 A game board is shown in the diagram. Players take turns to roll an ordinary fair die, then move their counters forward from ‘start’ a number of squares equal to the number rolled with the die. If a player’s counter ends its move on a coloured square, then it is moved back to the start 18 17 15 14 13 10 11 a Find the probability that a player’s counter is on ‘start’ after rolling the die: i once ii twice. b Find the probability that after rolling the die three times, a player’s counter is on: i 18 ii 17 DID YOU KNOW? It has long been common practice to write or philosophical argument is complete. demonstrandum, meaning ‘which is what had to be shown’....Q E D. at the point where a mathematical proof.Q E D. is an |
initialism of the Latin phrase quad erat Latin was used as the language of international communication, scholarship and science until well into the 18th century. Q.E.D. does not stand for Quite Easily Done! A popular modern alternative is to write 5W, an abbreviation of Which Was What Was Wanted. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability Application of the multiplication law The multiplication law given in Key point 4.8 can be used to show whether or not events are independent. If we can show that A B independent, and vice versa. If, for example, XP( independent only if X Y ) P( × and YP( ∩ = 0.3= ) ), then A and B are 0.4= ) P(. 0.12 B ) 0.4 0.3 P( P( A ) ∩ = × =, then X and Y are WORKED EXAMPLE 4.7 Events,J K and L are independent. Given that P( J ) = 0.5, P( K ) = 0.6 and P( J L ) ∩ = 0.24, find: a P( )∩J K b P( )L c P( )∩K L. Answer a P( J K∩ ) = 0.5 × 0.6 = 0.3 b 0.5 P( × L ) = 0.24 P( L ) = = 0.24 0.5 0.48 c P( K L ) ∩ = = 0.48 0.6 × 0.288 We know that P( J K ∩ ) P( = J ) P( × K. ) We know that P( J ) P( × L ) P( = J L ∩. ) We know that K L P( ∩ = ) P( K ) P( × L. ) 105 Examples of independence and non-independence that you may have come across are: ● Enjoyment of sport is independent of gender if equal proportions of males and females enjoy sport. ● If unequal proportions of employed and unemployed people own cars, then car ownership is not independent of employment status – it is dependent on it. WORKED EXAMPLE 4 |
.8 In a group of 60 students, 27 are male ( )H. The )M and 20 study History ( Venn diagram shows the numbers of students in these and other categories. One student is selected at random from the group. Show that the events ‘a male is selected’ and ‘a student who studies History is selected’ are independent. 60 27 M H 18 9 11 20 22 Answer Does M P( ) P( × H ) P( = M H ) ∩? If the multiplication law holds for the events M and H, then they are independent. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy TIP When we show that two events are independent (or not), it is important to state a conclusion in words after doing the mathematics. A short sentence like the last line of Worked example 4.8 is sufficient. Writing.Q E D., however, is optional.. Cambridge International AS & A Level Mathematics: Probability & Statistics 1 P( M ) = 27 60, P( H ) = 20 60 and P( M H ∩ ) = 9 60 We state the three probabilities concerned. P( M ) P( × H ) = = = 27 60 9 60 P( M H ∩ ) × 20 60 P( Then we evaluate M show whether or not it is equal to P( )∩M H. ) P( × H to ) The multiplication law holds for events M and H, therefore they are independent...Q E D. EXPLORE 4.2 In Worked example 4.8, we showed that the events M and H are independent. For the 60 students in that particular group, we could show by similar methods whether or not the three pairs of events M and independent. ′M and H and ′M and ′H, ′H are Do this then discuss what you believe would be the most appropriate conclusion to write. 106 EXERCISE 4D 1 Y and Z are independent events. YP( ) 0.7= and ZP( ) 0.9=. Find P( )∩Y Z. 2 Two independent events are M and N. Given that MP( )N., find P( 0.21 M N ∩ P |
( = ) ) = 0.75 and 3 Independent events S and T are such that P( 0.4=S ) and P( ) ′ =T 0.2. Find: a P( )∩S T b P( ′ ∩S T. ) 4,A B and C are independent events, and it is given that A B∩ = P( ) 0.35, 5, P( B C∩ = ) 0.56 and P( A C ) ∩ = 0.4. a Express P( )A in terms of: i P( )B ii P( )C. b Use your answers to part a to find: i P( )B ii P( )′A iii P( B C. ) ′ ∩ ′ 5 In a class of 28 children, 19 attend drama classes, 13 attend singing lessons, and six attend both drama classes and singing lessons. One child is chosen at random from the class. Event D is ‘a child who attends drama classes is chosen’. Event S is ‘a child who attends singing lessons is chosen’. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability a Illustrate the data in an appropriate table or diagram. b Are events D and S independent? Give a reason for your answer. 6 Each child in a group of 80 was asked )R or )M. The results whether they regularly read ( regularly watch a movie ( are given in the Venn diagram opposite. One child is selected at random from the group. Event R is ‘a child who regularly reads is selected’ and event M is ‘a child who regularly watches a movie is selected’. 80 R M 12 20 30 18 107 Determine, with justification, whether events R and M are independent. P 7 Two fair 4-sided dice, both with faces marked 1, 2, 3 and 4, are rolled. Event A is ‘the sum of the numbers obtained is a prime number’. Event B is ‘the product of the numbers obtained is an even number’. a Find, in simplest form, the value of P( )A, of P( )B and of P( ) |
∩A B. b Determine, with justification, whether events A and B are independent. c Give a reason why events A and B are not mutually exclusive. 8 Two ordinary fair dice are rolled. Event X is ‘the product of the two numbers obtained is odd’. Event Y is ‘the sum of the two numbers obtained is a multiple of 3’. a Determine, giving reasons for your answer, whether X and Y are independent. b Are events X and Y mutually exclusive? Justify your answer. 9 A fair 8-sided die has faces marked 1, 2, 3, 4, 5, 6, 7 and 8. The score when the die is rolled is the number on the face that the die lands on. The die is rolled twice. Event V is ‘one of the scores is exactly 4 less than the other score’. Event W is ‘the product of the scores is less than 13’. Determine whether events V and W are independent, justifying your answer. PS 10 Two hundred children are categorised by gender and by whether or not they own a bicycle. Of the 108 males, 60 own a bicycle, and altogether 90 children do not own a bicycle. a Tabulate these data. b Determine, giving reasons for your answer, whether ownership of a bicycle is independent of gender for these 200 children. c What percentage of the females and what percentage of the males own bicycles? Explain how your answers to part c confirm the result obtained in part b. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 PS 11 At an election, it was found that people voted for Party X independently of their income group. The following table shows that 12 400 people from three income groups voted altogether, and that 7440 of them voted for Party X. Low Medium High Totals Party X Totals a 3100 b 6820 c 2480 7440 12 400 Find the value of a, of b and of c. PS 12 The speed limit at a motorway junction is 120 km/h. Information about the speeds and directions in which 207 vehicles were being driven are shown in the following table. North East South Under limit Over limit 36 15 27 |
15 36 18 West 39 21 Providing evidence to support your answer, determine which vehicles’ speeds were independent of their direction of travel. FAST FORWARD We will study discrete random variables that arise from independent events in Chapter 6. 4.4 Conditional probability The word conditional is used to describe a probability that is dependent on some additional information given about an outcome or event. 108 For example, if your friend randomly selects a letter from the word ACE, then P(selects E) =. 1 3 However, if we are told that she selects a vowel, we now have a conditional probability that is not the same as P(selects E). This conditional probability is P(selects E, given that she selects a vowel) =. 1 2 Conditional probabilities are usually written using the symbol | to mean given that. We read P( )A B as ‘the probability that A occurs, given that B occurs’. | WORKED EXAMPLE 4.9 A child is selected at random from a group of 11 boys and nine girls, and one of the girls is called Rose. Find the probability that Rose is selected, given that a girl is selected. Answer P(Rose is selected | a girl is selected) = 1 9 The additional information, ‘given that a girl is selected’, reduces the number of possible selections from 20 to 9, and Rose is one of those nine girls. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability WORKED EXAMPLE 4.10 The following table shows the numbers of students in a class who study Biology ( )B and who study Chemistry ( )C. B ′B Totals Represent the data in a suitable Venn diagram, and find the probability that a randomly selected student: 9 7 16 8 1 9 C ′C Totals Answer 25 B 7 9 8 17 8 25 C 1 a P( | C B ) = 9 16 b P WORKED EXAMPLE 4.11 a studies Chemistry, given that they study Biology b does not study Biology, given that they do not study Chemistry. The number of students in each category is shown in the Venn diagram. 16 study Biology and nine of these study Chemistry. Eight students do not study Chemistry and one of |
these does not study Biology. 109 Two children are selected at random from a group of five boys and seven girls. Find the probability that the second child selected is a boy, given that the first child selected is: a a boy b a girl. Answer a P(second is a boy | first is a boy) = b P(second is a boy | first is a girl) = 4 11 5 11 If a boy is selected first, then the second child is selected from four boys and seven girls. If a girl is selected first, then the second child is selected from five boys and six girls. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 EXERCISE 4E 1 One letter is randomly selected from the six letters in the word BANANA. Find the probability that: a an N is selected, given that an A is not selected b an A is selected, given that an N is not selected. 2 One hundred children were each asked whether they have brothers )S. Their responses are given in )B and whether they have sisters ( ( the Venn diagram opposite. 100 B S Find the probability that a randomly selected child has: 16 48 24 a sisters, given that they have brothers b brothers, given that they do not have sisters c sisters or brothers, given that they do not have both. 12 3 Two photographs are randomly selected from a pack of 12 colour and eight black and white photographs. Find the probability that the second photograph selected is colour, given that the first is: a colour b black and white. 8 4 3 1 5 D 11 2 H 4 The Venn diagram opposite shows the responses of 40 girls who were )D asked if they have an interest in a career in nursing ( or human rights ( )N, dentistry ( )H. 40 N 110 a Find the probability that a randomly selected girl has an interest in: i human rights, given that she has an interest in nursing ii nursing, given that she has no interest in dentistry. b Describe any group of girls for whom dentistry is the least popular 6 career of interest. 5 The quiz marks of 40 students are represented in the following bar chart 10 5 Marks Two students are selected at random from the group. Find |
the probability that the second student: a scored more than 5, given that the first student did not score more than 5 b scored more than 7, given that the first student scored more than 7. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 6 The histogram shown represents the times taken, in minutes, for 115 men to complete a task. Two men are selected at random from the group. Find the probability that the: a first man took less than 1 minute, given that he took less than 3 minutes b second man took less than 6 minutes, given that the first man took less than 1 minute 20 15 10 5 0 7 At an insurance company, 60% of the staff are male ( )M and )FT. The following Venn diagram shows 70% work full-time ( this and one other piece of information. Chapter 4: Probability 1 2 3 4 5 6 7 8 9 10 Time taken (min) 1 M FT a What information is given by the value 0.10 in the Venn 0.60 a b c 0.70 diagram? b Find the value of a, of b and of c. c An employee is randomly selected. Find: i P( M FT ) | ii P( | FT M )′ [ iii P ( ) | ( M FT M FT ∪ ∩ 0.10 ) ] 8 Two fair triangular spinners, both with sides marked 1, 2 and 3, are spun. Given that the sum of the two numbers spun is even, find the probability that the two numbers are the same. 9 Two ordinary fair dice are rolled and the two numbers rolled are added together to give the score. Given that a player’s score is greater than 6, find the probability that it is not greater than 8. 111 10 The circular archery target shown, on which 1, 2, 3 or 5 points can be scored, is divided into four parts of unequal area by concentric circles. The radii of the circles are 3cm, 9cm, 15cm and 30 cm. You may assume that a randomly fired arrow pierces just one of the four areas and is equally likely to pierce any part of the target. a Show that the probability of scoring 5 points is 0.01. b Find the probability of |
scoring 3 points, 2 points and 1 point with an arrow. c Given that an arrow does not score 5 points, find the probability that it scores 1 point. d Given that a total score of 6 points is obtained with two randomly fired arrows, find the probability that neither arrow scores 1 point. 1 2 3 5 Independence and conditional probability At the beginning of Section 4.3, ‘independent’ was described in quite familiar terms. In general, we can use the multiplication law given in Key point 4.8 as the definition of ‘independent’. However, a more formal definition can now be given. Events X and Y are said to be independent if each is unaffected by the occurrence of the other. If this is the case then the probability that X occurs is the same in two complementary situations: (i) when Y occurs, and (ii) when Y does not occur. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 From these, we can now say that X and Y are independent if and only if X Y P( | ) P( = | X Y ) ′. Consider rolling an ordinary fair die and the events X and Y, as defined below. X: the outcome is a square number (1 or 4). Y: the outcome is an odd number (1, 3 or 5). First note that 1 is odd and square, so X and Y are not mutually exclusive; but are they independent? When Y occurs, the die shows 1, 3 or 5, so P( | X Y ) = 1 3 When Y does not occur, the die shows 2, 4 or 6, so P( X ) = 1 3, whether Y occurs or not. | X Y ) P( = P( Events X and Y are not mutually exclusive, but they are independent. ) ′ means that P(X) is unaffected by the occurrence of Y. | X Y EXPLORE 4.3 1 Consider rolling an ordinary fair die. In each case below, determine whether the given events are mutually exclusive, and whether they are independent. a b ‘X: a number less than 3’, and ‘Y: an even number’. ‘A: |
a number that is 4 or more’, and ‘B: a number that is not more than 4’. 112 2 Now consider rolling a fair 12-sided die, numbered from 1 to 12. In each case below, determine whether the given events are mutually exclusive, and whether they are independent. a b c ‘X: an even number’, and ‘Y: a factor of 28’. ‘A: a prime number’, and ‘B: a multiple of 3’. ‘F: a factor of 12’, and ‘M: a multiple of 5’. 3 An integer from 1 and 20 inclusive is selected at random. Three events are defined as follows: A: the number is a multiple of 3. B: the number is a factor of 72. C: the number has exactly two digits, and at least one of those digits is a 1. Determine which of the three possible pairs of these events is independent. 4.5 Dependent events and conditional probability Two events are mutually dependent when neither can occur without being affected by the occurrence of the other. An example of this is when we make selections without replacement; that is, when probabilities for the second selection depend on the outcome of the first selection. The multiplication law for independent events (see Key point 4.8) is a special case of the multiplication law of probability. The multiplication law of probability is used to find the probability that ‘this and that’ occurs when the events involved might not be independent. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy KEY POINT 4.9 Chapter 4: Probability P( and ) P( A B = P( and ) P( B A ∩ = ) P( A ) P( × ) | B A B ) P( × | A B ) P( A ) P( × | B A ) P( ≡ B ) P( × A B. | ) WORKED EXAMPLE 4.12 Two children are randomly selected from 11 boys ( consists of: )B and 14 girls ( )G. Find the probability that the selection a two boys b a boy and a girl, in any order. Answer a P(2 boys) P( |
and ) ( × ) P( B × 1 11 10 24 25 11 60 b P (a boy and a girl) P= (B and G) P+ (G and B) ) P( + ) P( × ) P( = ) | G B 1 2 14 25 + × 14 24 × 11 24 B 1 11 25 77 150 = = The first child is selected from 25 but the second is selected from the remaining 24. This tells us that the selections are dependent and that we must use conditional probabilities (suffices mean 1st and 2nd). There are two different orders in which a boy and a girl can be selected, but note that P(B and G) = P(G and B). 113 WORKED EXAMPLE 4.13 Every Saturday, a man invites his sister to the theatre or to the cinema. 70% of his invitations are to the theatre and 90% of these are accepted. His sister rejects 40% of his invitations to the cinema. Find the probability that the brother’s invitation is accepted on any particular Saturday. Answer The probability that the sister accepts depends on where she is invited to go. This tells us that our calculations must involve conditional probabilities. 0.7 0.3 T C 0.9 0.1 0.6 0.4 accepts.... P(T and accepts) = 0.7 × 0.9 = 0.63 rejects..... P(T and rejects) = 0.7 × 0.1 = 0.07 accepts.... P(C and accepts) = 0.3 × 0.6 = 0.18 The given information is shown in the tree diagram, where T and C stand for ‘theatre’ and ‘cinema’. rejects..... P(C and rejects) = 0.3 × 0.4 = 0.12 P(accepts) P= (T and accepts) P+ (C and accepts) = = 0.63 0.18 + 0.81 The sister can accept an invitation to the theatre or to the cinema. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics |
: Probability & Statistics 1 We can use the multiplication law of probability to find conditional probabilities. We know that A B P( P( )A are known. ∩ = ) P( A ) P( ×, so P( ) | B A B A can be found when P( ) | )∩A B and KEY POINT 4.10 P( ) | B A = ) P( A B ∩ ) P( A and P( | A B ) = WORKED EXAMPLE 4.14, where A B P( ∩ ≡ ) P( ∩. ) B A TIP The symbol ≡ means ‘is identical to’. P( ) B A ∩ P( B ) Given that A B ) ∩ = P( 0.36 and BP( ) 0.9=, find P(A | B). Answer P( P( ) B A ∩ P( B A B ∩ P( B 0.36 0.9 0.4 ) 114 WORKED EXAMPLE 4.15 An ordinary fair die is rolled. Find the probability that the number obtained is prime, given that it is odd. Answer P(prime | odd) = P(odd and prime) P(odd WORKED EXAMPLE 4.16 The odd numbers are 1, 3 and 5, so 3 6 P(odd) = are 3 and 5, so P(odd and prime). The odd prime numbers 2 =. 6 TIP Alternatively, two of the three odd numbers on a die are prime, so P(prime | odd) 2 =. 3 A boy walks to school (W) 60% of the time and cycles (C) 40% of the time. He is late to school (L), on 5% of the occasions that he walks, and he is late on 2% of the occasions that he cycles. Given that he is late to school, find the probability that he cycles; that is, find P( | C L ). Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Answer 0.6 W 0.4 C 0.05 0.95 0.02 L...... P(W and L) = 0.030 Lʹ..... P(W and L� |
�) = 0.570 L..... P(C and L) = 0.008 0.98 Lʹ..... P(C and Lʹ) = 0.392 P( L ) P( = W L and ) P( and ) L C + 0.008 = = 0.030 + 0.038 Chapter 4: Probability The tree diagram shows the given information. The boy can arrive late by walking or by cycling. P( and ) L C P( L 0.008 0.038 4 19 or 0.211 EXERCISE 4F 115 1 Two ties are taken at random from a bag of three plain and five striped ties. By use of a tree diagram, or otherwise, find the probability that both ties are: a plain b striped. 2 There are four toffee sweets and seven nutty sweets in a girl’s pocket. Find the probability that two sweets, selected at random, one after the other, are not the same type. 3 On a library shelf there are seven novels, three dictionaries and two atlases. Two books are randomly selected without replacement from these. Find the probability that the selected books are: a both novels b both dictionaries or both atlases. 4 A woman travels to work by bicycle 70% of the time and by scooter 30% of the time. If she uses her bicycle she is late 3% of the time but if she uses her scooter she is late only 2% of the time. a Find the probability that the woman is late for work on any particular day. b Given that the woman expects not to be late on approximately 223 days in a year, find the number of days in a year on which she works. 5 Two children are randomly selected from a group of five boys and seven girls. Determine which is more likely to be selected: a b two boys or two girls? the two youngest girls or the two oldest boys? Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 6 A boy has five different pairs of shoes mixed up under his bed. Find the probability that when he selects two shoes at random they can be worn as a matching pair. 7 A bag contains five 4cm |
nails, six 7 cm nails and nine 10 cm nails. Find the probability that two randomly selected nails: a have a total length of 14cm b are both 7 cm long, given that they have a total length of 14cm. 8 Yvonne and Novac play two games of tennis every Saturday. Yvonne has a 65% chance of winning the first game and, if she wins it, her chances of winning the second game increase to 70%. However, if she loses the first game, then her chances of winning the second game decrease to 55%. Find the probability that Yvonne: a loses the second game b wins the first game, given that she loses the second game. 9 a Given X Y ) ∩ = P( 0.13 and XP( ) = 0.65, find P( Y X. | ) b Given M N P( ∩ ) = c Given P( V W ∩ ) = 0.27 and N MP( | ) = 0.81, find P( )M. 0.35 and P( ) ′ =W 0.60, find P( V W. | ) 116 10 When a customer at a furniture store makes a purchase, there is a 15% chance that they purchase a bed. Given that 4.2% of all customers at the store purchase a bed, find the probability that a customer does not make a purchase at the store. 11 A number between 10 and 100 inclusive is selected at random. Find the probability that the number is a multiple of 5, given that none of its digits is a 5. 12 Three of Mr Jumbillo’s seven children, who include one set of twins, are selected at random. a Calculate the probability that exactly one of the twins is selected. b Given that exactly three of the children are girls, find the probability that the selection of three children contains more girls than boys. 13 Anya calls Zara once each evening before she goes to bed. She calls Zara’s mobile phone with probability 0.8 or her landline. The probability that Zara answers her mobile phone is 0.74, and the probability that she answers her landline is y. This information is displayed in the tree diagram shown. 0.8 Calls mobile Calls landline 0.74 Answers Does not answer y Answers Does not answer a Given that Zara answers 68% of Anya’s calls, find the value of y. b Given that Anya’s |
call is not answered, find the probability that it is made to Zara’s landline. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability FAST FORWARD We will study further techniques for calculating probabilities using permutations and combinations in Chapter 5, Section 5.4. 117 14 Every Friday, Arif offers to take his sons to the beach or to the park. The sons refuse an offer to the beach with probability 0.65 and accept an offer to the park with probability 0.85. The probability that Arif offers to take them to the beach is x. This information is shown in the tree diagram. a Find the value of x, given that 33% of Arif’s offers are refused. Beach x Park Accept 0.65 Refuse 0.85 Accept Refuse b Given that Arif’s offer is accepted, find the probability that he offers to take his sons to the park. PS 15 Two children are selected from a group in which there are 10 more boys than girls. Given that there are 756 equiprobable ordered selections that can occur, find the probability that two boys or two girls are selected. PS PS 16 There is a 43% chance that Riya meets her friend Jasmine when she travels to work. Given that Riya walks to work and does not meet Jasmine 30% of the time, and that she travels to work by a different method and meets Jasmine 25% of the time, find the probability that Riya walks to work on any particular day. 17 Aaliyah buys a randomly selected magazine that contains a crossword puzzle on five randomly chosen days of each week. On 84% of the occasions that she buys a magazine, she attempts its crossword, which she manages to complete 60% of the time. Find the probability that, on any particular day, Aaliyah does not complete the crossword in a magazine. Checklist of learning and understanding ● Probabilities are assigned on a scale from 0 (impossible) to 1 (certain). ● When one object is randomly selected from n objects, P(selecting any particular object) 1 =. n ● P(event) = Number of favourable equally likely outcomes Total number of equally likely outcomes ● P( A ) |
P(not + A ) = or A P( 1 ) P( + A ) ′ = 1 ● In n trials, event A is expected to occur n × AP( times. ) ● ∪A B means ‘A or B’ and ∩A B means ‘A and B’. ● Mutually exclusive events have no common favourable outcomes. For mutually exclusive events A and B, P( or A B ) P( = A B ∪ ) P( = A ) P( + B ) ● Non-mutually exclusive events have at least one common favourable outcome. For any two events A and B, P( or A B ) P( = A B ∪ ) P( = A ) P( + B ) P( − A B ∩ ) ● Independent events can occur without being affected by the occurrence of each other. Events A and B are independent if and only if A P(A | B) = P(A | B ′). P( and ) P( B = A B ∩ ) P( = A ) P( × B ), such that ● For any two events A and B, A P( and ) P( B = A B ∩ ) P( = A ) P( × | B A P( and B A ) | ) = P( A B ∩ P( A ) ). Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 END-OF-CHAPTER REVIEW EXERCISE 4 1 Four quality-control officers were asked to test 1214 randomly selected electronic components from a company’s production line, and to report the proportions that they found to be defective. The proportions reported were: 2 333, 3 411, 0 187 and 4 283. These figures confirm what the manager thought; that about %k of the components produced are defective. Of the 7150 components that will be produced next month, approximately how many does the manager expect to be defective? [2] 2 Three referees are needed at an international tournament and there are 12 to choose from: three from Bosnia, four from Chad and five from Denmark. If the referees are selected at random, find the probability that at least two of |
them are from the same country. [3] 3 The diagram opposite gives details about a company’s 115 Part-time Full-time employees. For example, it employs four unqualified, part-time females. Male 10 Unqualified 42 Two employees are selected at random. Find the probability that: a one is a qualified male and the other is an unqualified female [2] b both are unqualified, given that neither is employed part-time. [3] 7 4 3 2 Female 16 33 4 The numbers of books read in the past 3 months by the members of a reading club are shown in the following table. 118 No. books No. members f( ) 2< 3 2– 4 5 5–7 22 8–9 7 10 2 10> 1 Find the probability that three randomly selected members have all read fewer than eight books, given that they have all read more than four books. [3] 5 When they are switched on, certain small devices independently produce outputs of 1, 2 or 3 volts with respective probabilities of 0.3, 0.6 and 0.1. Find the probability that three of these devices produce an output with a sum of 5 or 6 volts. [4] 6 One hundred people are attending a conference. The following Venn diagram shows how many are male ( )M, have brown eyes ( )BE and are right-handed ( )RH. M BE 100 47 39 RH 55 87 a Given that there are 43 males with brown eyes, 42 right-handed males and 46 right-handed people with brown eyes, copy and complete the Venn diagram. b Two attendees are selected at random. Find the probability that: i they are both females who are not right-handed ii exactly one of them is right-handed, given that neither of them have brown eyes. [3] [2] [3] Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 4: Probability 7 A student travels to college by either of two routes, A or B. The probability that they use route A is 0.3, and the probability that they are passed by a bus on their way to college on any particular day is 0.034. They are twice as likely to be passed by a bus when they use route B as when |
they use route A. a Use the tree diagram opposite to form and solve a pair of simultaneous equations in x and y. b Find the probability that the student uses route B, given that they are not passed by a bus on their way to college. [3] [3] x y 0.3 0.7 A B 8 Two ordinary fair dice are rolled. If the first shows a number less than 3, then the score is the mean of the numbers obtained; otherwise the score is equal to half the absolute (non-negative) difference between the numbers obtained. Find the probability that the score is: a positive b greater than 1, given that it is less than 2 c less than 2, given that it is greater than 1. Bus No bus Bus No bus [3] [3] [3] 9 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, A B C, • The probability that Rick chooses entrance A is 1 3, are all equal. and D. Each friend chooses an entrance independently.. The probabilities that he chooses entrances B C, or D • Brenda is equally likely to choose any of the four entrances. • The probability that Ali chooses entrance C is 2 7 The probabilities that he chooses the other two entrances are equal. and the probability that he chooses entrance D is 3 5 i Find the probability that at least 2 friends will choose entrance B. ii Find the probability that the three friends will all choose the same entrance. 119. [4] [4] Cambridge International AS & A Level Mathematics 9709 Paper 61 Q5 November 2010 10 Maria chooses toast for her breakfast with probability 0.85. If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8. If she has a bread roll then the probability that she will have jam on it is 0.4. i Draw a fully labelled tree diagram to show this information. ii Given that Maria did not have jam for breakfast, find the probability that she had toast. [2] [4] Cambridge International AS & A Level Mathematics 9709 Paper 62 Q3 November 2009 11 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either ‘low’, ‘medium’ or ‘high’. The table shows |
the number of countries in each category. Birth rate Low Medium High Low GDP Medium High 3 20 35 5 42 8 45 12 0 Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 One of these countries is chosen at random. i Find the probability that the country chosen has a medium GDP. ii Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP. iii State with a reason whether or not the events ‘the country chosen has a high GDP’ and ‘the country chosen has a high birth rate’ are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate. [1] [2] [2] iv Find the probability that both countries chosen have a medium GDP and a medium birth rate. [3] Cambridge International AS & A Level Mathematics 9709 Paper 63 Q3 November 2012 12 Three boxes, A, B and C, each contain orange balls and blue balls, as shown. box A box B box C 120 probability that it is from box C. a A girl selects a ball at random from a randomly selected box. Given that she selects a blue ball, find the b A boy randomly selects one ball from each box. Given that he selects exactly one blue ball, find the probability that it is from box A. 13 A and B are independent events. If AP( ) = 0.45 and P( 14 Two ordinary fair dice are rolled. Event A is ‘the sum of the numbers rolled is 2, 3 or 4’. B = ) 0.64 P[(, find A B )∪ ′]. Event B is ‘the absolute difference between the numbers rolled is 2, 3 or 4’. a When the dice are rolled, they show a 1 and a 3. Explain why this result shows that events A and B are not mutually exclusive. b Explain how you know that A B P( ) ∩ = 1 18. c Determine whether events A and B are independent. 15 A and B are independent events, where A B P( ∩ ′ = ) 0 |
.14, A B ′ ∩ = P( ) 0.39 and P( A B ∩ < ) 0.25. Use a Venn diagram, or otherwise, to find A B P[( )∪ ′]. 16 In a survey, adults were asked to answer yes or no to the question ‘Do you regularly watch the evening TV news?’ Some of the results from the survey are detailed in the Venn diagram opposite. One adult is selected at random and it is found that the events ‘a female is selected’ and ‘a person who regularly watches the evening TV news is selected’ are independent. Find the number of adults questioned in the survey. [4] Male No 90 x 63 105 Copyright Material - Review Only - Not for Redistribution [3] [4] [2] [1] [2] [3] [4] Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 17 Bookings made at a hotel include a room plus any meal combination )L and supper ( of breakfast ( )S. The Venn diagram opposite shows the number of each type of booking made by 71 guests on Friday. a A guest who has not booked all three meals is selected at random. )B, lunch ( Find the probability that this guest: i has booked breakfast or supper [2] ii has not booked supper, given that they have booked lunch. [2] b Find the probability that two randomly selected guests have both booked lunch, given that they have both booked at least two meals. [3] Chapter 4: Probability 71 B L 24 6 3 7 5 16 1 9 S PS PS 18 Three strangers meet on a train. Assuming that a person is equally likely to be born in any of the 12 months of the year, find the probability that at least two of these three people were born in the same month of the year. 19 A box contains three black and four white chess pieces. Find the probability that a random selection of five chess pieces, taken one at a time without replacement, contains exactly two black pieces which are selected one immediately after the other. PS 20 The following table shows the numbers of IGCSE (I) and A Level (A) examinations passed by a group of university students. IGCSEs (I) 5 13 10 |
0 0 0 1 A Levels (A) a For a student selected at random, find: i P(I + A = 11 | A < 4) ii P(I A > 5 | I + A > 10) − b Six students who all have at least three A Level passes are selected at random. Find the greatest possible range of the total number of IGCSE passes that they could have. 121 [4] [4] [2] [3] [2] Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy 122 Chapter 5 Permutations and combinations In this chapter you will learn how to: ■ solve simple problems involving selections ■ solve problems about arrangements of objects in a line, including those involving repetition and ■ evaluate probabilities by calculation using permutations or combinations. restriction Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 5: Permutations and combinations PREREQUISITE KNOWLEDGE Where it comes from What you should be able to do Check your skills Chapter 4, Section 4.4. Recognise and calculate conditional probabilities. An experiment has 10 equally likely outcomes: three are favourable to event A, five are favourable to event B and four are favourable to neither A nor B. P( Find A B ) and B A P( ). | | Simple situations with millions of possibilities This topic is concerned with selections and arrangements of objects. Permutations and combinations appear in many complex modern applications: transport logistics; relationships between proteins in genetic engineering; sorting algorithms in computer science; and protecting computer passwords and e-commerce transactions in cryptography. A selection of objects is called a combination if the order of selection does not matter; however, if the order of selection does matter, then the selection is called a permutation. To make the difference between a permutation and a combination clear, we can describe them as follows. ● A combination is a way of selecting objects. There are three combinations of two letters from A, B and C. These are A and B, A and C, ● A permutation is a way of selecting objects and arranging |
them in a particular order. There are six permutations of two letters from A, B and C. These are and B and C. AB, BA, AC, CA, BC and CB. EXPLORE 5.1 123 Each letter A to Z is encrypted (or transformed) to a fixed distinct letter using its position in the alphabet (A 1, B 2, C 3, = the password SATURN is encrypted as ECHKBP. = …. By doing this, = ) This information gives us six clues to work out the method of encryption (e.g. S 3→, and so on). 1 E→ means 19 5→, A C→ means An Enigma Encryption Machine, circa 1940. Investigate the method of encryption and then find the password that is encrypted as UJSNOL. There are over 27 million possible passwords, so the probability that a random guess is correct is approximately zero. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 5.1 The factorial function In this chapter, we will frequently need to write and evaluate expressions such as 4 × × ×. 3 2 1 A shorthand method of doing this is to use the factorial function: 4 factorial’ and is written 4! On most calculators, the factorial function appears as n! or x! × × × is called ‘four 3 2 1 As examples: 7! 6! 4 5040 = × = 30 5! − 4! = 4!(5 1) − = 4! 4 × = 96 3! 0 KEY POINT 5.1! n = ( n n – 1)( – 2)… 3 2 1, for any integer × × × n n 0>. 0! 1= The following figure shows values of n! in sequence, where the next term is obtained by division. From this it is clear that 0! must be equal to 1. 124 n! 7! 6! 5! 4! 3! 2 ÷ value 5040 720 120 24 6 2 1! 1 1 ÷ 0! 1 EXERCISE 5A 1 Without using a calculator, find the value of: a 5! 3! b 4! 2! 3!− |
c 7 × 4! 21 3! + × d 10! 8! + 9! 7! e 20! 18! − 13! 11! 2 Use your calculator to find the smallest value of n for which: a n! 1000 000 > b 5! 6! × < n! c n(!)! 1020 > 3 Use your calculator to find the largest value of n for which: a n! 500 000 80< b 1.5 10 × 12 –! 0 n > 4 Express, in as many different ways as possible, the numbers 144, 252 and 1 1 a b, or c is equal to 0 or to 1. 5 Express the area of a 53cm by 52 cm rectangle using factorials. c n − a 2 in the form n (! 2)!! ×! c < 500 b!, where none of 6 Two cubical boxes measure 25cm by 24cm by 23cm, and 8cm by 7 cm by 6cm. Express the difference between their volumes using factorials. 7 Eight children each have seven boxes of six eggs and each egg is worth $0.09. Write the total value of all these eggs in dollars, using factorials. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 5: Permutations and combinations DID YOU KNOW? There is a famous legend about the Grand Vizier in Persia who invented chess. The King was so delighted with the new game that he invited the Vizier to name his own reward. The Vizier replied that, being a modest man, he wanted only one grain of wheat on the first square of a chessboard, two grains on the second, four on the third, and so on, with twice as many grains on each square as on the previous square. The innumerate King agreed, not realising that the total number of grains on all 64 squares would be 2 – 1 wheat production for the next 150 years. 64, or 1.84 1019 ×, which is equivalent to the world’s present Although the number 2 64 – 1 is extremely large, it is only about one-third of 21! factorial. As a challenge, try showing that 2 0 1 2 + + 2 2 + … + 63 2 64 = 2 – 1 without using |
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