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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

The speeds, in kmh–1, of vehicles passing a particular point on a rural road are normally distributed with mean μ and standard deviation 20. Find the value of μ and find what percentage of the
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
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.
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
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
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
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
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
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
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
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
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
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) =
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
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
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
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
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
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.
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
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.
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
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
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
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 =
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,
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.
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
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
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
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
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
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
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
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
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 >
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
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
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
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.
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.
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
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
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
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.
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
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
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
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
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)
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
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%.
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
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.
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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).

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