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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... |
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... |
. 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,... |
) 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) = 12... |
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 ... |
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... |
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... |
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... |
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 vecto... |
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... |
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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 Cha... |
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 fa... |
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... |
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... |
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 Describ... |
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 t... |
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 ● F... |
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 num... |
� = [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 F... |
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 num... |
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 ... |
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 - Cambrid... |
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... |
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 Unive... |
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 ... |
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 ... |
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 Th... |
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 Tw... |
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 ... |
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 t... |
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 de... |
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 writt... |
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. Ide... |
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 - ... |
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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, w... |
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 stan... |
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} indivi... |
�. 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 ... |
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 val... |
, 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 de... |
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 T... |
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 o... |
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 1... |
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... |
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 vari... |
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 ... |
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 e... |
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 tempe... |
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 ... |
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 ... |
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 greates... |
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 ... |
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. F... |
. 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 ave... |
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. corre... |
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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 ... |
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 ab... |
: 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 ... |
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 ... |
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 th... |
) 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 ‘fa... |
� 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 w... |
� = ) 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 inters... |
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 ... |
)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 a... |
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 electri... |
: 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 Co... |
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 2... |
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, wi... |
+ + = (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... |
: 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 i... |
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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 - ... |
.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 1... |
( = ) ) = 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 ... |
∩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... |
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 pro... |
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) ... |
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 - Cam... |
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 condition... |
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... |
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... |
: 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( ∩... |
�) = 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... |
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 w... |
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 - Cam... |
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 exclusiv... |
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 a... |
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 numbe... |
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 Pres... |
.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 selecte... |
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] Copyri... |
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 E... |
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... |
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