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a formula. 5.2 Permutations We can make a permutation by taking a number of objects and arranging them in a line. For example, the two possible permutations of the digits 5 and 9 are the numbers 59 and 95. Although there are several methods that we can use to find the number of possible permutations of objects, all me... |
choices). Finally, we place the remaining digit at the right side, as shown in the following diagram. 3 choices × 2 choices × 1 choices The numbers above the lines in the diagram are not the digits that are being arranged – they are the numbers of choices that we have for placing the three digits. The three digits can... |
in the middle. If her youngest child sits on the adjacent chair to her left, in how many ways can the remaining children be seated? PS 7 A group of n boys can be arranged in a line in a certain number of ways. By adding two more boys to the group, the number of possible arrangements increases by a factor of 420. Find ... |
three Us, two As and two Gs. In the formula of Key point 5.3, excluding 1! for the D in the denominator does not change our answer. EXERCISE 5C 1 Find the number of distinct arrangements of all the letters in these words: a TABLE b TABLET c COMMITTEE d MISSISSIPPI e HULLABALLOO. 2 Find how many six-digit numbers can b... |
72 1011 numbers in our daily lives, so we are likely to see this as just a very large number whose true size we cannot really comprehend until it is put into some human context. For example, if everyone on Earth over the age of 14 (i.e. about 5.46 109 contributed one new arrangement of the word every day starting on 1s... |
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distinguishable and the mangoes: ) and three watermelons W( ) can be placed in a must not be separated b must be separated Answer a 2P2 M1 M2 W1 W2 W3 1 object 4P4 2 P 2 × 4 P 4 = 48 ways The two mangoes can be placed next to each other in P2 2 ways. This pair is now considered as a single object to be arranged with t... |
how many of the six-digit numbers that can be made from 1, 2, 2, 3, 3 and 3: a begin with a 2 b are not divisible by 2. 7 Find the number of distinct arrangements that can be made from all the letters in the word THEATRE when the arrangement: a begins with two Ts and ends with two Es b has H as its middle letter c end... |
be made from the seven digits 3, 4, 5, 6, 7, 8 and 9, if each is used at most once? Answer KEY POINT 5.4 n ( = There are P r! n n r − permutations of r objects from n distinct objects. )! 7 P 3 = 7! (7 3)! − 7! 4! 7 6 5 = × × 210 = = three-digit numbers We select and arrange just three of the seven distinct digits (an... |
: a five from seven distinct objects b four from nine distinct objects. 2 From 12 books, how many ways are there to select and arrange exactly half of them in a row on a shelf? 3 In how many ways can gold, silver and bronze medals be awarded for first, second and third places in a race between 20 athletes? You may assu... |
in terms of n and r. 12 Five playing cards are randomly selected from a standard deck of 52 cards. These five cards are shuffled, and then the top three cards are placed in a row on a table. How many different arrangements of three of the 52 cards are possible? PS 13 Seven chairs, A to G, are arranged as shown. C D E ... |
we call this is a combination. KEY POINT 5.5 A combination of r objects which are then arranged in order is equivalent to a permutation. We write nCr to mean the number of combinations of r objects from n. Since there are rPr = r! ways of arranging the r objects, we have!( r n − ( )! r )! Suppose we wish to select thr... |
Level Mathematics: Probability & Statistics 1 WORKED EXAMPLE 5.13 A team of five is to be chosen from six women and five men. Find the number of possible teams in which there will be more women than men. Answer From 6 women From 5 men No. teams 3 4 5 or or = 200 6 C 4 × 5 C 75 = 200 + 75 6 + = 281 teams with more wome... |
x is a positive integer. 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 - Ca... |
has 10 different posters to pin up in their classroom but there is enough space for only five of them. They have three posters on algebra, two on calculus and five on trigonometry. In how many ways can they choose the five posters to pin up if: a b c there are no restrictions they decide not to pin up either of the ca... |
them is flawed. Which of the two answers is correct? Can you explain why the other answer is not correct? 5.4 Problem solving with permutations and combinations Permutations and combinations can be used to find probabilities for certain events. If an event consists of a number of favourable permutations that are equip... |
Answer From 13 red From 7 black Number of ways 5 4 3 or or 0 1 2 13 C 5 × 7 C 1287 = 0 13 C 4 × 7 C 5005 = 1 13 C 3 × 7 C 2 = 6006 Total 12298 = 20 C 15504 5 = ways P(more red than black) = 12298 15504 or 0.793. The table shows the possible make-up of the selected cherries when there are more red than black; and also ... |
itesides) 1 P(same side The events ‘sitting on the same side’ and ‘sitting on opposite sides’ are complementary. 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... |
to test six randomly selected animals. Find the probability that the selection consists of: a equal numbers of cattle and sheep b more females than males. 10 a How many distinct arrangements of the letters in the word STATISTICS are there? b Find the probability that a randomly selected arrangement begins with: i thre... |
�. 142 a Investigate the value of XP( b Hence, express the value of for any value of n ⩾ 2? ) for values of n from 2 to 5. P( P( X ′ X ) ) in terms of n. Can you justify your answer WEB LINK You will find a range of interesting and challenging probability problems (with hints and solutions) in Module 16 on the NRICH we... |
tall; four are 20 cm tall and three are 25cm tall. Find the number of ways that the books can be arranged on the shelf so that none of them is shorter than the book directly to its right. 6 The 11 letters of the word REMEMBRANCE are arranged in a line. i Find the number of different arrangements if there are no restri... |
paired up, playing each other once with the losing team being eliminated. How many games are played during the whole tournament? [3 11 A bank provides each account holder with a nine-digit card number that is arranged in three blocks, as shown in the example opposite. Find, in index form, the number of card numbers av... |
] [2] [3] [4] [5] [2] [4] [4] [4] [3] [6] 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 Cha... |
least possible value of Z X−, given that Y 25=. c Given that none of the players drew any of their games and that X Z 50, find the exact mean number − = of games won by the players. 2 Six books are randomly given to two girls so that each receives at least one book. a In how many ways can this be done? [1] [1] [2] [3]... |
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 2 b Two roles in the play must be played by girls; three roles must be played by boys, but the fool can be played ... |
Find the probability that none of them has a university degree. 13 One hundred qualified drivers are selected at random. Out of these 100 drivers, of the 40 drivers who wear spectacles, 30 passed their driving test at the first attempt. Altogether, 25 of the drivers did not pass at their first attempt. a Show the data... |
.6 million vehicle owners in the country, and that each owns, on average, 1.183 registered vehicles. [3] 17 Seats for the guests at an awards ceremony are arranged in two rows of eight and ten, divided by an aisle, as shown. e l s i a Seats are randomly allocated to 18 guests. a Find the probability that two particular... |
Tools of the trade Suppose a trading company is planning a new marketing campaign. The campaign will probably go ahead only if the most likely outcome is that sales will increase. However, the company also needs to be aware of worst-case and best-case outcomes, as sales may decrease or decrease dramatically, stay the ... |
2 and 3, we write X ∈ {1, 2, 3}, where the symbol ∈ means ‘is an element of’. REWIND We learnt how to find probabilities for selections with and without replacement in Chapters 4 and 5, Sections 4.3, 4.4, 4.5 and 5.4. Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Rev... |
the probability distribution for the random variable V. v P( )==V v 2 0.05 3 2c 4 5 6 0.1+c 2 +c 0.05 0.16 Find the value of the constant c and find P( V >. 4) Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Rev... |
15 Selecting one from Y8, and two from Y2 ′. P( Y = 2) = 8 C × 2 10 C 2 C 1 3 = 7 15 P( Y = 3) = 8 C × 3 10 C 2 C 0 3 = 7 15 y P( )==Y y 1 1 15 2 7 15 3 7 15 Selecting two from Y8, and one from Y2 ′. Selecting three from Y8, and none from Y2 ′. The table shows the probability distribution for Y. TIP Always check that ... |
�s six trucks, five vans, three cars and one motorbike are randomly selected and tested for roadworthiness. a Show that the probability of selecting three vans is 2. 91 b Draw up the probability distribution table for the number of vans selected. c Find the probability that, at most, one van is selected. 7 Five grapes ... |
y, find the value of the constant k.. 13 Q is a discrete random variable and Q {3, 4, 5, 6} ∈ a Given that Q q P( = ) = cq 2, find the value of the constant c. b Hence, find QP( 4)>. 14 Four books are randomly selected from a box containing 10 novels, 10 reference books and 5 dictionaries. The random variable N represe... |
than 9. 18 The discrete random variable R is such that R {1, 3, 5, 7}. 1) ( k r + 2 r + a Given that R r, find the value of the constant k. P( ∈ ) = = 4)øR. b Hence, find P( EXPLORE 6.1 Consider the probability distribution for X, the number of heads obtained when two fair coins are tossed, which was given in the tabl... |
the greatest mathematicians for 350 years. has no positive integer solutions for any integer = + y z n n The theorem is simple in that it says ‘a square can be divided into two squares, but a cube cannot be divided into two cubes, nor a fourth power into two fourth powers, and so on’. Pierre de Fermat himself claimed ... |
as being the long-term average value of X over a large number of trials. 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 Un... |
of the spread of values around the mean, XE( using probabilities in place of frequencies. ). These measures, like XE( ), can be calculated If we replace f by p and replace x by XE( ) in the second of the two formulae for variance, we obtain Σ 2 x p p Σ − X{E( KEY POINT 6.3 )} 2, which simplifies to Σ 2 x p − X{E( )} 2... |
)= P( into the formula ) E( X = Σ xp. Substitute into the formula for Var(X). Take the square root of the variance. EXERCISE 6B 158 1 The probability distribution for the random variable X is given in the following table. x 0 1 2 3 P( )==X x 0.10 0.12 0.36 0.42 Calculate XE( ) and XVar( ). 2 The probability distributi... |
this case. b Investigate the expected outcome and the standard deviation when the grading is reversed (i.e. high profit is graded 1, and so on). Compare these outcomes with those from part a. 8 Two ordinary fair dice are rolled. The discrete random variable X is the lowest common multiple of the two numbers rolled. a ... |
$1000 loan. b Assuming that the company charges the fee found in part a, how would it be possible, without changing the loan conditions, for the company’s expected profit from each $1000 loan to be greater than 40%? 13 When a scout group of 8 juniors and 12 seniors meets on a Monday evening, one scout is randomly sele... |
AST FORWARD We will study the expectation of two special discrete random variables, and the variance of one of them, in Chapter 7, Sections 7.1 and 7.2. Copyright Material - Review Only - Not for Redistribution Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy... |
10 0.4 q 0.2 101 0.2 a Given that YVar( ) 1385.2 =, show that q 2 – 61 q + 624 = and solve this equation. 0 b Find the greatest possible value of YE( ). [4] [2] 3 An investment company has produced the following table, which shows the probabilities of various percentage profits on money invested over a period of 3 yea... |
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 6: Probability distributions 7 Two ordinary fair dice are rolled. The product and the sum of the two numbers obtained are calc... |
Y is such that Y {4, 5, 8, 14, 17} ∈ )= and Y y P( is directly proportional to Find YP( 4)>. 12 X is a discrete random variable and X {0, 1, 2, 3}, find ∈. 0)> 0 or 2) øX 0.62 XP( P( 2 | X = =. Given that XP( > 1) = 0.24, P(0 < X < 3) = 0.5 and 1. 1+ y [4] [5] 13 Four students are to be selected at random from a group... |
[2] 164 Cambridge International AS & A Level Mathematics 9709 Paper 62 Q5 November 2009 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 C... |
used to model the number of successes in a fixed number • A geometric distribution can be used to model the number of trials up to and including the first success in an infinite number of independent trials. 7.1 The binomial distribution Consider an experiment in which we roll four ordinary fair dice. In each independ... |
�� 1 6 2 3 4 2 1 0 5 6 5 6 5 6 4 Cr. These probabilities are the terms in the 4 (6666) In the table, we see that P( R r = ) = 1 + binomial expansion of 6 5 6 4. notation, the five probabilities shown in the previous table are given by Using the R r = P( = ) n r ... |
IP n is For work involving binomial expansions, the notation Cr rarely used nowadays. Your calculator may use this notation but it has mostly been 167 replaced by n r . REWIND We met a series of independent events with just two possible outcomes in the Explore 6.3 activity in Chapter 6, Section 6.3. TIP Coeffi... |
~ B(8, 0.7), find P( X > 6), correct to 3 significant figures. Given that X Answer P( X > 6) P( = X = 7) P( + X = 8) X ~ B(8, 0.7) tells us that = = = 8 7 0.7 × 0.197650 8 1 0.3 + × 0.057648 … 8 7 × … + 0.7 8 0 0.3 × q p = = 8,, 0.3 0.7, n = and that. X {0, 1, 2, 3, 4, 5, 6, 7, 8} ∈ 0.255 Copyright... |
0.15 0 × … + 0.001502 ] … Let the random variable X be the number in the sample with R+ blood, then X. ~ B(40, 0.85) REWIND Recall from Chapter 4, Section 4.1 that P( A ′ 1 P( − ) A ). = EXPLORE 7.1 Binomial distributions can be investigated using the Binomial Distribution resource on the GeoGebra website. We co... |
Copy Cambridge International AS & A Level Mathematics: Probability & Statistics 1 3… = 0.6 0.216; 0.6 4 = 5 0.1296; 0.6 = 0.07776. The least possible value is =n 5. Alternatively, we can solve 0.6 n < 0.1 by trial and improvement. We know that n is an integer, so we 3 1 evaluate 0.6, 0.6, 0.6, 2 … up to the first one ... |
are married. 10 A footballer has a 95% chance of scoring each penalty kick that she takes. Find the probability that she: a b scores from all of her next 10 penalty kicks fails to score from exactly one of her next seven penalty kicks. 11 On average, 13% of all tomato seeds of a particular variety fail to germinate wi... |
< 1) 0.006, find the least possible value of n. M 19 The number of damaged eggs, D, in cartons of six eggs have been recorded by an inspector at a packing depot. The following table shows the frequency distribution of some of the numbers of damaged eggs in 150 000 boxes. 171 No. damaged eggs ( )D 0 1 No. cartons ( )f ... |
suffers from colour-blindness. DID YOU KNOW? Although Pascal’s triangle is named after the 17th century French thinker Blaise Pascal, it was known about in China and in Persia as early as the 11th century. The earliest surviving display is of Jia Xian’s triangle in a work compiled in 1261 by Yang Hui, as shown in the ... |
= 2 x p ) – {E( Var( X = Σ Our experiment consists of should not be surprised to find that E( = × n = trials with a probability of success 2 = × 0.6 1.2 = np.16) 0.48) (0 (2 X 0.36) – 1.2 p = 2 )} 2 2 (1 2 2 = 0.6 0.48 in each, so we REWIND We saw in Chapter 4, Section 4.1 that event A is expected to occur times. ) P(... |
) X = 7.5, find: We use q = npq np = ) Var( X ) E( X to find p. E( ) =X np, so =n ) E( X p 0.375 11 × 0.625 21 X ~ B(32, 0.375) 32 11 × 0.138 b P( X = 11) = = EXERCISE 7B 1 Calculate the expectation, variance and standard deviation of each of the following discrete random 174 variables. Give non-exact answer... |
- 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 7: The binomial and geometric distributions, and its standard deviation is one-third of its mean. Calcu... |
a series of repeated independent trials, is a discrete random variable whose distribution is called a geometric distribution. The following table shows the probability that the first success occurs on the rth trial. r P( )== X r 1 p 2 3 4 (1 – ) p p (1 – )2 p p (1 – )3 p p........ n (1 – ) p 1 p n−........ Copyright M... |
there is only one way to obtain the first success on the rth trial, and that REWIND We saw in Chapter 6, Section 6.2 that 1 Σ =p for a probability distribution. You will also have seen geometric progressions and geometric series in Pure Mathematics 1, Chapter 6. TIP r q –1 =, p X r = An alternative form of this formul... |
the first r trials) = These two results are written in terms of q in Key point 7.4. WORKED EXAMPLE 7.8 KEY POINT 7.4 ~ Geo( p ) p, then When and X 1 –=q =<P( X r • ) 1 – • P( )> X r = qr qr 177 In a particular country, 18% of adults wear contact lenses. Adults are randomly selected and interviewed one at a time. Find ... |
random variable X ~ Geo(0.2), find: a =XP( 7) b P( X ≠ 5) c >XP( 4). 2 Given that T ~ Geo(0.32), find: a =TP( 3) b øTP( 6) c >TP( 7). 3 The probability that Mike is shown a yellow card in any football match that he plays is probability that Mike is next shown a yellow card: 1 2. Find the a in the third match that he p... |
2) b =YP( 2) c P( X = 1 and Y = 1). 9 On average, 14% of the vehicles being driven along a stretch of road are heavy goods vehicles (HGVs). A girl stands on a footbridge above the road and counts the number of vehicles, up to and including the first HGV that passes. Find the probability that she counts: a at most thre... |
Mathematics: Probability & Statistics 1 PS 15 Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining: a b the first double on the fourth roll the first pair of numbers with a sum of more than 10 before the 10th roll. PS 16 X ~ Geo(0.24) and Y ~ Geo(0.25) are two independent random variable... |
we have X {1, 2, 3, 4,...} and ∈ =p x {,, p pq pq 2, 3 pq,...}. Step 1 of the proof is to form an equation that expresses µ in terms of p and q. To do this we use µ = Σ xpx. 1 p. REWIND We studied the expectation of a discrete random variable in Chapter 6, Section 6.3. There are three more steps required to complete t... |
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of the die. 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 Unive... |
) 0 0.40 1 0.60 y 0 P( = ) Y y 0.60 1 0.40 Investigate the probability distributions for X and Y in a best-of-three contest, where the first player to win two games wins the contest. Who gains the advantage as the number of games played in a contest increases? What evidence do you have to support your answer? PS How l... |
2 A family has booked a long holiday in Skragness, where the probability of rain on any particular day is 0.3. Find the probability that: a b the first day of rain is on the third day of their holiday it does not rain for the first 2 weeks of their holiday. [2] [1] [2] 3 One plastic robot is given away free inside eac... |
remember or to forget to switch off the headlights. Giving your answers in their simplest index form, find the probability that on the next 16 occasions that they park their car in the evening, they forget to switch off the headlights: a 14 more times than they remember to switch them off b at least 12 more times than... |
25> ~ B(, 0.4) and that X P( = n P( X n = – 1), express the constant k in terms of n, and PS 12 A book publisher has noted that, on average, one page in eight contains at least one spelling error, one page in five contains at least one punctuation error, and that these errors occur independently and at random. The pub... |
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 Mathema... |
and temperature and so on. A continuous random variable is not defined for specific values. Instead, it is defined over an interval of values. Consider the mass of an apple, denoted by X grams. Within the range of possible masses, X can take any value, such as 111.2233…, or 137.8642…, or 145.2897…, or …. The probabili... |
the columns of an equal-width interval histogram (preferably one displaying large amounts of data with many classes) to model the probability distribution of a set of continuous data. In the case of a random variable, such a curved graph represents a function, = f( and is called a probability density function, abbrevi... |
24<ø x 3 9 18 24 18 9 3 3 6<ø y 6 9<ø y 9 12<ø y 12 15<ø y 15 18<ø y 18 21<ø y 21 24<ø y 8 19 10 4 10 19 8 Discuss and describe the shapes of the three graphs. What feature do they have in common? Compare the measures of central tendency (averages) for w, x and y. 190 The normal curve The frequency distribution of x i... |
investigate the effect of altering the mean and/or standard deviation on the location and shape of a normal curve by visiting the Density Curve of Normal Distribution resource on the GeoGebra website. Note that the area under the curve is always equal to 1, whatever the values of µ and σ. EXERCISE 8A 1 The probability... |
variance 4 ml2, and the mean quantity of peach juice is 340 ml with standard deviation 4 ml. a Copy the diagram and sketch onto it the normal curve for the quantity of peach juice in the peach juice tins. b Describe the curves’ differences and similarities women men 160 180 Height (cm) apple juice 340 Volume (ml) 5 Th... |
extremely unlikely, but not impossible. 0 and The continuous random variable would, therefore, indicate that P(mass P(mass 0.2) 6) 0. < > > > [Incidentally, the greatest ever recorded mass of a pineapple is 8.28 kg!] The probability distribution of a continuous random variable is a mathematical function that provides ... |
table. Properties Half of the values are less than the mean. Half of the values are greater than the mean. Approximately 68.26% of the values lie within 1 standard deviation of the mean. Approximately 95.44% of the values lie within 2 standard deviations of the mean. Approximately 99.72% of the values lie within 3 sta... |
interval from 60 to 84. For C: 8013 out of 10 974 observations lie in the interval from 112 to 134. Investigate this information (using the previous table showing properties and probabilities for normal distributions) and comment on the statement ‘The distributions of A, B and C are all normal’. The standard normal va... |
values of z have been 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 - Cambr... |
5714 0.6103 0.6480 0.5359 0.5753 0.6141 0.6517 1 4 4 4 4 4 432 8765 9 ADD 8 8 8 8 7 12 12 12 12 11 16 16 16 15 15 20 20 20 19 19 24 24 24 23 22 28 28 28 27 26 32 32 32 31 30 36 36 36 35 34 TIP Critical values refer to probabilities of 75%, 90%, 95%, … and their complements 25%, 10%, 5%, … and so on. 196 We locate the f... |
Given that Z ~ N(0, 1), find P(0.4 ø Z < 1.7) correct to 3 decimal places. Answer 0.4 1.7 z Φ (1.7) = 0.9554 and Φ (0.4) = 0.6554 P(0.4 ø < Z 1.7) = P( 1.7) – P( Z < 0.4) Z < (1.7) – = Φ Φ (0.4) The required probability is equal to the difference between the area to the left of z left of z and the area to the, as illu... |
P( ø –0.11) = 1 – 0.5438 = 0.4562. 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 ... |
. z( )Φ For our value of 0.9072, we need to add 0.0006 to 0.9066, so ‘ADD 6’ is required – this will be done if 1.32 is given a 4th figure of 4. We can check the value obtained for a by reading the tables in the usual way. (1.324) Φ ‘ADD 6’ (1.32) = Φ + = = 0.9066 + 0.0006 0.9072 Alternatively, we can show our working ... |
7) –0.013) 2 The random variable Z is normally distributed with mean 0 and variance 1. Find the following probabilities, correct to 3 significant figures. a d g j P(1.5 < Z < 2.5) P(–2.807 < Z < –1.282) P(–1.2 ( P 2 < Z < ø <Z 1.2) ) 5 b e h P(0.046 < Z < 1.272) c P(1.645 < Z < 2.326) P(–1.777 < Z < –0.746) P(–1.667 < ... |
) = 0.9516 P(–0.674 < Z c < ) = 0.725 P(–2.7 < Z c < ) = 0.0252 Standardising a normal distribution The probability distribution of a normally distributed random variable is represented by a normal curve. This curve is centred on the mean µ; the area under the curve is equal to 1, and its height is determined by the st... |
or multiplication affects the mean and the standard deviation. FAST FORWARD We will learn more about coding random variables in the Probability & Statistics 2 Coursebook, Chapter 3. REWIND In the table showing properties and probabilities of normal distributions prior to Explore 8.3, we saw that probabilities are deter... |
0.4633 Area to the left of z = Φ (–1.342) (0) – Φ Total area X< ∴ P(2 = = 0.4633 < 9) = 0= is [ 0.5 – 1 – 0.4102 0.4102 + 0.874 0.5 Φ (1.342) ] = 0.8735 Here, we find the two areas separately then add them to obtain our final answer, which is where we round to the accuracy specified in the question. TIP Where possible... |
Press - Review Copy Chapter 8: The normal distribution We subtract equation 1[ ] from 2[ ] to solve this pair of simultaneous equations. [2] [1]. 7.78 12 – µ = 1.282 σ 10 – µ = 0.674 σ 2 ∴σ = = 3.29 0.608 σ and µ = EXERCISE 8C 1 Standardise the appropriate value(s) of the normal variable X represented in each diagram,... |
(6.2 P(26 Find P(8 10), given that X ~ N(12, 2.56). 3 a Find a, given that X ~ N(30, 16) and that P( X a< ) = 0.8944. b Find b, given that X ~ N(12, 4) and that P( X b< ) = 0.9599. c Find c, given that X ~ N(23, 9) and that P( X c > ) = 0.9332. d Find d, given that X ~ N(17, 25) and that P( X d > ) = 0.0951. e Find e, ... |
� < Q P(4 P( Q. 5) < 1.288) = 0.281 and P( Q < 6.472) = 0.591, find the value of µ 12 For the variable ~ N(, V )2µ σ, it is given that P( V = 0.7509 and P( V > 9.2) = 0.1385. Find the value of µ and of σ, and calculate VP( 8.4) < 10)ø. 13 Find the value of µ and of σ and calculate 0.6858 = and WP( 4.75) 2.25) WP( ø ù =... |
�� (0.512) = 0.6957 We standardise the mass of 3.5 kg. 0.6957 is a relative frequency equal to 69.57% < P(mass 69.57% of 1356 ∴ There were about 943 newborn babies. 0.6957 = 943.3692 3.5 kg) = WORKED EXAMPLE 8.11 A factory produces half-litre tins of oil. The volume of oil in a tin is normally distributed with me... |
bolt is less than 18.85 cm long. 2 The waiting times, in minutes, for patients at a clinic are normally distributed with mean 13 and variance 16. a Calculate the probability that a randomly selected patient has to wait for more than 16.5 minutes. b Last month 468 patients attended the clinic. Calculate an estimate of ... |
Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Review Copy - Cambridge University Press - Review Copy Chapter 8: The normal distribution 9 The ages of the children in a gymnastics club are normally distributed with mean 15.2 years and standard deviation σ.... |
90 15 The time taken in seconds for Ginger’s computer to open a specific large document is normally distributed with mean 9 and variance 5.91. a Find the probability that it takes exactly 5 seconds or more to open the document. b Ginger opens the document on her computer on n occasions. The probability that it fails to... |
, and 12 tosses of a fair coin are shown in the following diagrams. Notice that, as the number of coin tosses increases, the shape of the probability distribution becomes increasingly normal. 0.5 p = 0.5, n = 2 0.4 p = 0.5, n = 4 0.25 p = 0.5, n = 12 208 10 12 Does the binomial probability distribution maintain its nor... |
.25 p = 0.95, q = 0.05 0 25 0 25 0 25 0 25 np = 3.75, nq = 21.25 np = 8.75, nq = 16.25 np = 18.75, nq = 6.25 np = 23.75, nq = 1.25 As you can see, the binomial distribution loses its normal shape when p is small and also when q is small. KEY POINT 8.4 ) ~ B(, n p can be X approximated by )2µ σ, where N(, 2σ = npq, µ = ... |
( 13), where X = 13 is included, we calculate using X. = 13.5 Further details of continuity corrections are given in Worked example 8.12. WORKED EXAMPLE 8.12 Given that ~ B(100, 0.4) X to find: a P( X < 43) b P( X > 43), use a suitable approximation and continuity correction Copyright Material - Review Only - Not for R... |
, Chapter 2. TIP X a< means ‘X is fewer/less than a’. X a> means ‘X is more/greater than a’. X a< means ‘X is at most a’ and ‘X is not more than a’ and ‘X is a or less’. X a> means ‘X is at least a’ and ‘X is not less than a’ and ‘X is a or more’. WORKED EXAMPLE 8.13 Boxes are packed with 8000 randomly selected items. ... |
make the continuity correction! TIP Although our answer to part a is only an approximation, we should not use a rounded probability, such as 0.8, in further calculations. 211 WORKED EXAMPLE 8.14 A fair coin is tossed 888 times. Find, by use of a suitable approximation, the probability that the coin lands heads-up at m... |
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