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mathematics
cambridge international as & a level mathematics probability & statistics
Cambridge International AS & A Level Mathematics Probability & Statistics 1 Coursebook 1st Edition Dean Chalmers, Julian Gilbey - Solutions
In a certain town, 63% of homes have an internet connection.a. In a random sample of 20 homes in this town, find the probability that:i. Exactly 15 have an internet connectionii. Exactly nine do not have an internet connection.b. Use a suitable approximation to find the probability that more than
The masses, in kilograms, of ‘giant Botswana cabbages’ have a normal distribution with mean μ and standard deviation 0.75. It is given that 35.2% of the cabbages have a mass of less than 3kg. Find the value of μ and the percentage of cabbages with masses of less than 3.5kg.
For the variable V ~N(μ, σ2), it is given that P(V < 8.4) = 0.7509 and P(V > 9.2) = 0.1385. Find the value of μ and of σ, and calculate P(V ≤ 10).
Coffee beans are packed into bags by the workers on a farm, and each bag claims to contain 200 g. The actual mass of coffee beans in a bag is normally distributed with mean 210 g and standard deviation σ . The farm owner informs the workers that they must repack any bag containing less than 200 g
17% of the people interviewed in a survey said they watch more than two hours of TV per day. A random sample of 300 of those who were interviewed is taken. Find an approximate value for the probability that at least one-fifth of those in the sample watch more than two hours of TV per day.
The ages of the vehicles owned by a large fleet-hire company are normally distributed with mean 43 months and standard deviation σ. The probability that a randomly chosen vehicle is more than 4 1/6 years old is 0.28. Find what percentage of the company’s vehicles are less than two years old.
Find the value of μ and of σ and calculate P(W > 6.48) for the variable W ~N(μ, σ2), given that P(W ≥ 4.75) = 0.6858 and P(W ≤ 2.25) = 0.0489.
Colleen exercises at home every day. The length of time she does this is normally distributed with mean 12.8 minutes and standard deviation σ. She exercises for more than 15 minutes on 42 days in a year of 365 days.a. Calculate the value of σ.b. On how many days in a year would you expect Colleen
An opinion poll was taken before an election. The table shows the percentage of voters who said they would vote for parties A, B and C.ind an approximation for the probability that, in a random sample of 120 of these voters:a. Exactly 50 said they would vote for party Bb. More than 70 but fewer
The weights, X grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.i. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.ii. Find the value of k such that P(k < X < 128) = 0.7465.iii. Five bars of soap are chosen
Find the value of μ and of σ and calculate P(W > 6.48) for the variable W ~N(μ, σ2), given that P(W ≥ 4.75) = 0.6858 and P(W ≤ 2.25) = 0.0489.
The times taken by 15-year-olds to solve a certain puzzle are normally distributed with mean μ and standard deviation 7.42 minutes.a. Find the value of μ, given that three-quarters of all 15-year-olds take over 20 minutes to solve the puzzle.b. Calculate an estimate of the value of n, given that
Boxes containing 24 floor tiles are loaded into vans for distribution. In a load of 80 boxes there are, on average, three damaged floor tiles. Find, approximately, the probability that:a. There are more than 65 damaged tiles in a load of 1600 boxesb. In five loads, each containing 1600 boxes,
Crates of tea should contain 200 kg, but it is known that 1 out of 45 crates, on average, is underweight. A sample of 630 crates is selected at random.a. Find the probability that more than 12 but fewer than 17 crates are underweight.b. Given that more than 12 but fewer than 17 crates are
X has a normal distribution, such that P(X > 147.0) = 0.0136 and P(X ≤ 59.0) = 0.0038. Use this information to calculate the probability that 80.0 ≤ X < 130.0.
Once a week, Haziq rows his boat from the island where he lives to the mainland. The journey time, X minutes, is normally distributed with mean μ and variance σ2.a. Given that P(20 ≤ X < 30) = 0.32 and that P(X < 20) = 0.63, find the values of μ and σ2.b. The time taken for Haziq to row
The lengths, Xcm, of the leaves of a particular species of tree are normally distributed with mean μ and variance σ2.a. Find P(μ – σ ≤ X < μ + σ).b. Find the probability that a randomly selected leaf from this species has a length that is more than 2 standard deviations from the
It is known that 2% of the cheapest memory sticks on the market are defective.a. In a random sample of 400 of these memory sticks, find approximately the probability that at least five but at most 11 are defective.b. Ten samples of 400 memory sticks are tested. Find an approximate value for the
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
Randomly selected members of the public were asked whether they approved of plans to build a new sports centre and 57% said they approved. Find approximately the probability that more than 75 out of 120 people said they approved, given that at least 60 said they approved.
On average, 8% of the candidates sitting an examination are awarded a merit. Groups of 50 candidates are selected at random.a. How many candidates in each group are not expected to be awarded a merit?b. Calculate the variance of the number of merits in the groups of 50.c. Find the probability
The time taken, T seconds, to open a graphics programme on a computer is normally distributed with mean 20 and standard deviation σ.Given that P(T > 13 | T ≤ 27) = 0.8, find the value of:a. σb. k for which P(T > k) = 0.75.
On average, 13% of all tomato seeds of a particular variety fail to germinate within 10 days of planting. Find the probability that 34 or 35 out of 40 randomly selected seeds succeed in germinating within 10 days of planting.
Decide whether or not it would be appropriate to model the distribution of X by a geometric distribution in the following situations. In those cases for which it is not appropriate, give a reason.a. A bag contains two red sweets and many more green sweets. A child selects a sweet at random and eats
Given that X ~ B(n, 0.4) and that P(X = 1) = k × P(X = n – 1), express the constant k in terms of n, and find the smallest value of n for which k > 25.
The random variable T has a geometric distribution and it is given that P(T = 2)/P(T = 5) = 15.625. Find P(T = 3).
There is a 15% chance of rain on any particular day during the next 14 days. Find the probability that, during the next 14 days, it rains on:a. Exactly 2 days b. At most 2 days.
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 publisher checks 480 randomly selected pages from various books for errors.a.
A factory makes electronic circuit boards and, on average, 0.3% of them have a minor fault. Find the probability that a random sample of 200 circuit boards contains:a. Exactly one with a minor fault b. Fewer than two with a minor fault.
Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining:a. The first double on the fourth rollb. The first pair of numbers with a sum of more than 10 before the 10th roll.
Given that Q~B(n, 0.3) and that P(Q = 0) > 0.1, find the greatest possible value of n.
Given that R~B(n, 0.8) and that P(R > n −1)<0.006, find the least possible value of n.
The number of months during the 4-month monsoon season (June to September) in which the total rainfall was greater than 5 metres, R, has been recorded at a location in Meghalaya for the past 32 years, and is shown in the following table.The distribution of R is to be modelled by R~B(4, p).a. Find
A mixed hockey team consists of five men and six women. The heights of individual men are denoted by hm metres and the heights of individual women are denoted by hw metres. It is given that Σhw = 9.84, Σhm = 9.08 and Σh2w =16.25.a. Calculate the mean height of the 11 team members.b. Given
Standardise the appropriate value(s) of the normal variable X represented in each diagram, and find the required probabilities correct to 3 significant figures.a. Find P(X ≤ 11), given that X ~N(8, 25).b. Find P(X < 69.1), given that X ~N(72, 11).c. Find P(3 < X < 7), given that X ~N(5,
Find the smallest possible value of n for which the following binomial distributions can be well approximated by a normal distribution.a. B(n, 0.024) b. B(n, 0.15) c. B(n, 0.52) d. B(n, 0.7)
X ~ Geo( p) and P(X = 2) = 0.2464. Given that p < 0.5, find P(X > 3).
Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.i. Find the probability that at least 2 of the 5 integers are less than or equal to 4. Robert now generates n random integers between 1 and 9 inclusive. The random variable X is the number of these n integers
X ~ Geo(0.24) and Y ~ Geo(0.25) are two independent random variables. Find the probability that X +Y = 4.
There is a 50% chance that a six-year-old child drops an ice cream that they are eating. Ice creams are given to 5 six-year-old children.a. Find the probability that exactly one ice cream is dropped.b. 45 six-year-old children are divided into nine groups of five and each child is given an ice
Given that X ~ Geo( p) and that P(X ≤ 4) = 2385/2401, find P(1 ≤ X < 4).
Anna, Bel and Chai take turns, in that order, at rolling an ordinary fair die. The first person to roll a 6 wins the game.Find the ratio P(Anna wins) : P(Bel wins) : P(Chai wins), giving your answer in its simplest form.
A coin is biased such that heads is three times as likely as tails on each toss. The coin is tossed 12 times. The variables H and T are, respectively, the number of heads and the number of tails obtained. Find the value of P(H = 7)/P(T = 7).
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 150000 boxes.The distribution of D is to be modelled by D~B(6, p).a. Estimate a suitable value
The variable T ~B(n, 0.96) and it is given that P(T = n) > 0.5. Find the greatest possible value of n.
In a particular country, 90% of both females and males drink tea. Of those who drink tea, 40% of the females and 60% of the males drink it with sugar. Find the probability that in a random selection of two females and two males:a. All four people drink teab. An equal number of females and males
It is estimated that 0.5% of all left-handed people and 0.4% of all right-handed people suffer from some form of colour-blindness. A random sample of 200 left-handed and 300 right-handed people is taken. Find the probability that there is exactly one person in the sample that suffers from colour
The probability distributions for A and B are represented in the diagram.Indicate whether each of the following statements is true or false.a. μΑ > μΒb. σΑ < σBc. A and B have the same range of values.d. σ2Α = σ2Βe. At least half of the values in B are greater than μΑ.f. At most
Given that Z ~N(0, 1), find the following probabilities correct to 3 significant figures.a. P(Z < 0.567)b. P(Z ≤ 2.468) c. P(Z > –1.53) d. P(Z ≥ – 0.077)e. P(Z > 0.817) f. P(Z ≥ 2.009) g. P(Z < –1.75) h. P(Z ≤ –0.013)i. P(Z < 1.96) j. P(Z
The length of a bolt produced by a machine is normally distributed with mean 18.5cm and variance 0.7 cm2. Find the probability that a randomly selected bolt is less than 18.85cm long.
Decide whether or not each of the following binomial distributions can be well-approximated by a normal distribution.For those that can, state the values of the parameters μ and σ2.For those that cannot, state the reason.a. B(20, 0.6)b. B(30, 0.95)c. B(40, 0.13)d. B(50, 0.06)
A continuous random variable, X, has a normal distribution with mean 8 and standard deviation σ. Given that P(X > 5) = 0.9772, find P(X < 9.5).
A and B are events such that P(A ∩ B') = 0.196, P(A' ∩ B) = 0.286 and P[(A ∪ B)'] = 0.364, as shown in the Venn diagram opposite.a. Find the value of x and state what it represents.b. Explain how you know that events A and B are not mutually exclusive.c. Show that events A and B are
The diagram shows normal curves for the probability distributions of P and Q, that each contain n values.a. Write down a statement comparing:i. σP and σQii. The median value for P and the median value for Qiii. The interquartile range for P and the interquartile range for Q.b. The datasets P and
The random variable Z is normally distributed with mean 0 and variance 1. Find the following probabilities, correct to 3 significant figures.a. P(1.5 < Z < 2.5)b. P(0.046 < Z < 1.272)c. P(1.645 < Z < 2.326)d. P(–2.807 < Z < –1.282) e. P(–1.777 < Z <
Calculate the required probabilities correct to 3 significant figures.a. Find P(X ≤ 9.7) and P(X > 9.7), given that X ~N(6.2, 6.25).b. Find P(X ≤ 5) and P(X > 5), given that X ~N(3, 49).c. Find P(X > 33.4) and P(X ≤ 33.4), given that X ~N(37, 4).d. Find P(X < 13.5) and P(X ≥
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 the number who
The variable Y is normally distributed. Given that 10σ = 3μ and P(Y < 10) = 0.75, find P(Y ≥ 6).
Meng buys a packet of nine different bracelets. She takes two for herself and then shares the remainder at random between her two best friends.a. How many ways are there for Meng to select two bracelets?b. If the two friends receive at least one bracelet each, find the probability that one friend
Probability distributions for the quantity of apple juice in 500 apple juice tins and for the quantity of peach juice in 500 peach juice tins are both represented by normal curves.The mean quantity of apple juice is 340 ml with variance 4ml2, and the mean quantity of peach juice is 340 ml with
Given that Z ~N(0, 1), find the value of k, given that:a. P(Z < k) = 0.9087b. P(Z < k) = 0.5442c. P(Z > k) = 0.2743d. P(Z > k) = 0.0298e. P(Z < k) = 0.25f. P(Z < k) = 0.3552 g. P(Z > k) = 0.9296 h. P(Z > k) = 0.648i. P(–k < Z < k) = 0.9128j. P(–k < Z
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, given that X ~N(100, 64) and
Tomatoes from a certain producer have masses which are normally distributed with mean 90 grams and standard deviation 17.7 grams. The tomatoes are sorted into three categories by mass, as follows:Small: under 80 g; Medium: 80 g to 104 g; Large: over
Describe the binomial distribution that can be approximated by the normal distribution N(14, 10.5).
In Scotland, in November, on average 80% of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.i. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).ii. Give a reason why the
Every Friday evening Sunil either cooks a meal for Mina or buys her a take-away meal. The probability that he buys a take-away meal is 0.24. If Sunil cooks the meal, the probability that Mina enjoys it is 0.75, and if he buys her a take-away meal, the probability that she does not enjoy it is x.
Find the value of c in each of the following where Z has a normal distribution with μ = 0 and σ2 = 1.a. P(c < Z < 1.638) = 0.2673 b. P(c < Z < 2.878) = 0.4968c. P(1 < Z < c) = 0.1408 d. P(0.109 < Z < c) = 0.35e. P(c < Z < 2) = 0.6687 f. P(c < Z <
a. Find f , given that X ~N(10, 7) and that P( f ≤ X < 13.3) = 0.1922.b. Find g, given that X ~N(45, 50) and that P(g ≤ X < 55) = 0.5486.c. Find h, given that X ~N(7, 2) and that P(8 ≤ X < h) = 0.216.d. Find j , given that X ~N(20, 11) and that P(j ≤ X < 22) = 0.5.
The heights, in metres, of the trees in a forest are normally distributed with mean μ and standard deviation 3.6. Given that 75% of the trees are less than 10m high, find the value of μ.
By first evaluating np and npq, use a suitable approximation and continuity correction to find P(X < 75) for the discrete random variable X ~B(100, 0.7).
A biased coin is four times as likely to land heads up compared with tails up. The coin is tossed k times so that the probability that it lands tails up on at least one occasion is greater than 99%. Find the least possible value of k.
Anouar and Zane play a game in which they take turns at tossing a fair coin. The first person to toss heads is the winner. Anouar tosses the coin first, and the probability that he wins the game is 0.51 + 0.53 + 0.55 + 0.57 +….a. Describe the sequence of results represented by the value 0.55 in
The probability that a woman can connect to her home Wi-Fi at each attempt is 0.44. Find the probability that she fails to connect until her fifth attempt.
It is estimated that 1.3% of the matches produced at a factory are damaged in some way. A household box contains 462 matches.a. Calculate the expected number of damaged matches in a household box.b. Find the variance of the number of damaged matches and the variance of the number of undamaged
A footballer has a 95% chance of scoring each penalty kick that she takes. Find the probability that she:a. Scores from all of her next 10 penalty kicksb. Fails to score from exactly one of her next seven penalty kicks.
In Restaurant Bijoux 13% of customers rated the food as ‘poor’, 22% of customers rated the food as ‘satisfactory’ and 65% rated it as ‘good’. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.i. Find the probability that more than 2 and fewer than 12 of
A study reports that a particular gene in 0.2% of all people is defective. X is the number of randomly selected people, up to and including the first person that has this defective gene. Given that P(X ≤ b) > 0.865, find E(X ) and find the smallest possible value of b.
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 three vehicles b.
The random variable H ~ B(192, p), and E(H) is 24 times the standard deviation of H. Calculate the value of p and find the value of k, given that P(H = 2) = k × 2–379.
In a particular country, 58% of the adult population is married. Find the probability that exactly 12 out of 20 randomly selected adults are married.
Gina has been observing students at a university. Her data indicate that 60% of the males and 70% of the females are wearing earphones at any given time. She decides to interview randomly selected students and to interview males and females alternately.a. Use Gina’s observation data to find the
A standard deck of 52 playing cards has an equal number of hearts, spades, clubs and diamonds. A deck is shuffled and a card is randomly selected. Let X be the number of cards selected, up to and including the first diamond.a. Given that X follows a geometric distribution, describe the way in which
Two independent random variables are X ~ Geo(0.3) and Y ~ Geo(0.7). Find:a. P(X = 2)b. P(Y = 2)c. P(X = 1andY = 1).
The variable Q ~ B(n, ¼), and its standard deviation is one-third of its mean. Calculate the non-zero value of n and find P(5 < Q < 8).
Research shows that the owners of 63% of all saloon cars are male. Find the probability that exactly 20 out of 30 randomly selected saloon cars are owned by:a. Malesb. Females.
When a certain driver parks their car in the evenings, they are equally likely to 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
Sylvie and Thierry are members of a choir. The probabilities that they can sing a perfect high C note on each attempt are 4/7 and 5/8, respectively.a. Who is expected to fail fewer times before singing a high C note for the first time?b. Find the probability that both Sylvie and Thierry succeed in
In a manufacturing process, the probability that an item is faulty is 0.07. Items from those produced are selected at random and tested.a. Find the probability that the first faulty item is:i. The 12th item tested ii. Not one of the first 10 items tested iii. One of the first eight items
Give a reason why a binomial distribution would not be a suitable model for the distribution of X in each of the following situations.a. X is the height of the tallest person selected when three people are randomly chosen from a group of 10.b. X is the number of girls selected when two children are
A driving test is passed by 70% of people at their first attempt. Find the probability that exactly five out of eight randomly selected people pass at their first attempt.
Four ordinary fair dice are rolled.a. In how many ways can the four numbers obtained have a sum of 22?b. Find the probability that the four numbers obtained have a sum of 22.c. The four dice are rolled on eight occasions. Find the probability that the four numbers obtained have a sum of 22 on at
A biased 4-sided die is numbered 1, 3, 5 and 7. The probability of obtaining each score is proportional to that score.a. Find the expected number of times that the die will be rolled, up to and including the roll on which he first non-prime number is obtained.b. Find the probability that the first
It is known that 80% of the customers at a DIY store own a discount card. Customers queuing at a checkout are asked if they own a discount card.a. Find the probability that the first customer who owns a discount card is:i. The third customer asked ii. Not one of the first four customers
W has a binomial distribution, where E(W) = 2.7 and Var(W) = 0.27. Find the values of n and p and use them to draw up the probability distribution table for W.
A man has five packets and each contains three brown sugar cubes and one white sugar cube. He randomly selects one cube from each packet. Find the probability that he selects exactly one brown sugar cube.
A computer generates random numbers using any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The numbers appear on the screen in blocks of five digits, such as 50119 26317 40068 ....... Find the probability that:a. There are no 7s in the first blockb. The first zero appears in the first blockc. The
Let X be the number of times an ordinary fair die is rolled, up to and including the roll on which the first 6 is obtained. Find E(X ) and evaluate P[X > E(X )].
The sides of a fair 5-sided spinner are marked 1, 1, 2, 3 and 4. It is spun until the first score of 1 is obtained. Find the probability that it is spun:a. Exactly twice b. At most five times c. At least eight times.
Given that G ~ B(n, p), E(G) 24 1/2 = and Var(G) 10 5/24 = , find:a. The parameters of the distribution of Gb. P(G = 20).
Find the probability that each of the following events occur.a. Exactly five heads are obtained when a fair coin is tossed nine times.b. Exactly two 6s are obtained with 11 rolls of a fair die.
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