Binomial distribution

A manufacturer makes components for the car industry. On average 1.5% of the components made are defective.

Using a suitable model, find the probability that in a random sample of 36 components there are

exactly 2 defective components,

more than 3 defective components. Once every six months, the manufacturer tests whether the proportion of defective components has changed using hypotheses H₀:p = 0.015 and H₁:p ≠ 0.015 A random sample of 260 components is taken and 8 are found to be defective.

Using a 5% level of significance, complete the test.

Xian rolls a fair die 10 times. The random variable X represents the number of times the die lands on a six.

Using a suitable distribution for X, find

P(X = 3)

P(X 3) Xian repeats this experiment each day for 60 days and records the number of days when X = 3

Find the probability that there were at least 12 days when X = 3

Find an estimate for the total number of sixes that Xian will roll during these 60 days.

Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

A machine fills packets with sweets and of the packets also contain a prize. The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable T represents the number of packets of sweets that contain a prize in each box.

State a condition needed for T to be modelled by B(40, A box is selected at random.

the probability that the box has exactly 6 packets containing a prize,

the probability that the box has fewer than 3 packets containing a prize. Kamil’s sweet shop buys 5 boxes of these sweets.

Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than A random sample of 110 packets is taken and 9 packets contain a prize.

Use a suitable test to assess Kamil’s claim. You should

• state your hypotheses clearly

• use a 5% level of significance

George throws a ball at a target 15 times. Each time George throws the ball, the probability of the ball hitting the target is 0.48 The random variable X represents the number of times George hits the target in 15 throws.

Find

P (X = 3)

P (X 5) George now throws the ball at the target 250 times.

Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

State one disadvantage of using quota sampling compared with simple random sampling. In a university 8% of students are members of the university dance club. A random sample of 36 students is taken from the university. The random variable X represents the number of these students who are members of the dance club.

Using a suitable model for X, find

P(X = 4)

P(X 7) Only 40% of the university dance club members can dance the tango.

Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.

Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango.

A company sells a variety of bars of chocolate including Pearchoc bars. In an attempt to increase the sales of Pearchoc bars, the company runs a promotion where any Pearchoc wrapper that has the word WINNER printed on the inside wins a prize. The company prints the word WINNER on 5% of Pearchoc bar wrappers. Pearchoc bars are sold in packets of 12

Find the probability that in a randomly selected packet of Pearchoc bars, the number of wrappers with the word WINNER printed on them will be

exactly one,

more than two.

A customer bought 4 packets of Pearchoc bars.

Find the probability that only 1 of the packets contained exactly 1 wrapper with the word WINNER printed on it.

Before the promotion, 4% of all bars of chocolate sold in a supermarket were Pearchoc bars. After the promotion, the company selects a random sample of 150 sales of bars of chocolate from the supermarket database and finds that 10 of these sales were Pearchoc bars.

Using a suitable approximation and stating your hypotheses clearly, test whether or not the company’s promotion has been successful. Use a 5% level of significance.

In a call centre, the number of telephone calls, X, received during any 10-minute period follows a Poisson distribution with mean 9

Find

P(X 5)

P(4 X 10) The length of a working day is 7 hours.

Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.

Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

A fair coin is spun 6 times and the random variable T represents the number of tails obtained.

Give two reasons why a binomial model would be a suitable distribution for modelling T.

Find P(T = 5)

Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is This second coin is spun 6 times.

Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

The time taken for a randomly selected person to complete a test is M minutes, where M ~ N (14, ²) Given that 10% of people take less than 12 minutes to complete the test,

find the value of

Graham selects 15 people at random.

Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test.

Jovanna takes a random sample of n people. Using a normal approximation, the probability that fewer than 9 of these n people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.

Find the value of n.

A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.

Number of cherries	0	1	2	3	4	5	6
Frequency	24	37	21	12	4	2	0

Calculate the mean and the variance of these data.

Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake.

The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains

exactly 2 cherries,

at least 1 cherry.

Rays fruit cakes are sold in packets of 5

Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places.

Twelve packets of Rays fruit cakes are selected at random.

Find the probability that exactly 3 packets contain more than 10 cherries.

In a region of the UK, 5% of people have red hair. In a random sample of size n, taken from this region, the expected number of people with red hair is 3

Calculate the value of n.

A random sample of 20 people is taken from this region. Find the probability that

exactly 4 of these people have red hair,

at least 4 of these people have red hair.

Patrick claims that Reddman people have a probability greater than 5% of having red hair. In a random sample of 50 Reddman people, 4 of them have red hair.

Stating your hypotheses clearly, test Patrick’s claim. Use a 1% level of significance.

A bag contains a large number of counters with one of the numbers 4, 6 or 8 written on each of them in the ratio 5:3:2 respectively. A random sample of 2 counters is taken from the bag.

List all the possible samples of size 2 that can be taken.

The random variable M represents the mean value of the 2 counters.

find the sampling distribution for M.

A sample of n sets of 2 counters is taken. The random variable Y represents the number of these n sets that have a mean of 8

Calculate the minimum value of n such that P(Y 1) 0.9

The proportion of houses in Radville which are unable to receive digital radio is 25%. In a survey of a random sample of 30 houses taken from Radville, the number, X, of houses which are unable to receive digital radio is recorded.

Find P(5 ≤ X 11)

A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.

Test, at the 10% level of significance, the radio company’s claim. State your hypotheses clearly.

The continuous random variable L represents the error, in metres, made when a machine cuts poles to a target length. The distribution of L is a continuous uniform distribution over the interval [0, 0.5]

Find P(L < 0.4).

Write down E(L).

Calculate Var(L).

A random sample of 30 poles cut by this machine is taken.

Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length.

When a new machine cuts poles to a target length, the error, X metres, is modelled by the cumulative distribution function F(x) where

Using this model, find P(X 0.4)

A random sample of 100 poles cut by this new machine is taken.

Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.

A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter μ. The number 2 is on 40% of the counters and the number 5 is on 25% of the counters. All the remaining counters have numbers greater than 5 on them. A random sample of 10 counters is taken from the bag.

State whether or not each of the following is a statistic

S = the sum of the numbers on the counters in the sample,

D = the difference between the highest number in the sample and μ,

F = the number of counters in the sample with a number 5 on them.

The random variable T represents the number of counters in a random sample of 10 with the number 2 on them.

Specify the sampling distribution of T.

The counters are selected one by one.

Find the probability that the third counter selected is the first counter with the number 2 on it.

Accidents occur randomly at a road junction at a rate of 18 every year. The random variable X represents the number of accidents at this road junction in the next 6 months.

Write down the distribution of X.

Find P(X 7).

Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).

Find the probability that there is at least one accident in exactly 4 of the next 6 months.

In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3 Find the probability that

exactly 4 customers join the queue in the next 10 minutes,

more than 10 customers join the queue in the next 20 minutes.

When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, T minutes, has a continuous uniform distribution over the interval [0, 5]. The random variable T is independent of the number of people joining the queue.

Find P(T > 3.5)

In a random sample of 5 customers, the random variable C represents the number of customers who took more than 3.5 minutes paying the assistant.

Find P(C 3)

Bethan has just reached the front of the queue and starts paying the assistant.

Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.

As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable X.

Suggest a suitable distribution for X.

Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable S represents the final score, in points, for an applicant who chooses answers to this test at random.

Show that S = 5 X – 20

Find E(S) and Var(S).

An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.

Find P(S 20)

Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4

Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1

Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.

On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.

Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95

A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 The call centre telephones 100 people every hour.

Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour.

A random variable X has the distribution B(12, p).

Given that p = 0.25 find

P(X < 5)

Given that P(X = 0) = 0.05, find the value of p to 3 decimal places.

Given that the variance of X is 1.92, find the possible values of p.

In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm, of a randomly chosen stick has a continuous uniform distribution over the interval [7, 10]. A stick is selected at random from the box.

Find the probability that the stick is shorter than 9.5 cm.

To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm.

Find the probability of winning a bag of sweets.

To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm.

Find the probability of winning a soft toy.

Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm.

Find the probability that Jim’s plank contains at most 3 defects.

Shivani buys 6 planks each of length 100 cm.

Find the probability that fewer than 2 of Shivani’s planks contain at most 3 defects.

Using a suitable approximation, estimate the probability that the total number of defects on Shivani’s 6 planks is less than 18.

A disease occurs in 3% of a population.

State any assumptions that are required to model the number of people with the disease in a random sample of size n as a binomial distribution.

Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.

Find the mean and variance of the number of people with the disease in a random sample of 100 people.

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.

The queuing time in minutes, X, of a customer at a post office is modelled by the probability density function

Using integration, find

the mean queuing time of a customer,

the probability that a customer will queue for more than 5 minutes.

Three independent customers shop at the post office.

Find the probability that at least 2 of the customers queue for more than 5 minutes.

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

exactly 3 of the games,

fewer than half of the games.

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

Calculate the mean and variance for the number of these 60 games that Bhim loses.

Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.

Using a 5% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is . The probability of rejection in either tail should be as close as possible to 0.025

Find the actual significance level of this test.

In the sample of 50 the actual number of faulty bolts was 8.

Comment on the company’s claim in the light of this value. Justify your answer.

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.

Test at the 1% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly.

A manufacturer supplies DVD players to retailers in batches of 20. It has 5% of the players returned because they are faulty.

Write down a suitable model for the distribution of the number of faulty DVD players in a batch.

Find the probability that a batch contains

no faulty DVD players,

more than 4 faulty DVD players.

Find the mean and variance of the number of faulty DVD players in a batch.

A factory produces components of which 1% are defective. The components are packed in boxes of 10. A box is selected at random.

Find the probability that the box contains exactly one defective component.

Find the probability that there are at least 2 defective components in the box.

Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.