Poisson distribution

Customers arrive at a bank at a rate of 10 customers per 5 minutes. John suggests that a Poisson distribution would be a suitable model to describe the number of customers arriving at the bank in a 1-minute interval.

Write down two assumptions that John has made.

Given that the Poisson distribution is a suitable model,

find the probability that in any given 1-minute interval

exactly 2 customers arrive,

more than 4 customers arrive.

When there is sufficient evidence that the rate of customers arriving at the bank is higher than 10 customers per 5 minutes, the manager must open another service till to avoid long queues. In a 4-minute interval, the manager observes 9 customers arriving.

Test, at the 5% significance level, whether or not the manager has sufficient evidence to open another service till. State your hypotheses clearly.

Using a 1% significance level, determine the smallest number of customers the manager must observe in a 1-minute interval in order to have sufficient evidence to open another service till.

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 company receives telephone calls at random at a mean rate of 2.5 per hour.

Find the probability that the company receives

at least 4 telephone calls in the next hour,

exactly 3 telephone calls in the next 15 minutes.

Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2

The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10

Test at the 5% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.

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 survey it is found that barn owls occur randomly at a rate of 9 per 1000 km².

Find the probability that in a randomly selected area of 1000 km² there are at least 10 barn owls.

Find the probability that in a randomly selected area of 200 km² there are exactly 2 barn owls.

Using a suitable approximation, find the probability that in a randomly selected area of 50000 km² there are at least 470 barn owls.

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.

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.

The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25 Find the probability that

a randomly chosen metre of cloth has 1 defect,

the total number of defects in a randomly chosen 6 metre length of cloth is more than 2

A tailor buys 300 metres of cloth.

Using a suitable approximation find the probability that the tailor’s cloth will contain less than 90 defects.

An online shop sells a computer game at an average rate of 1 per day.

Find the probability that the shop sells more than 10 games in a 7 day period.

Once every 7 days the shop has games delivered before it opens.

Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05

In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.

Using a suitable approximation and a 5% level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.

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.

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.

In a village, power cuts occur randomly at a rate of 3 per year.

Find the probability that in any given year there will be

exactly 7 power cuts,

at least 4 power cuts.

Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20

A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.

The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.

Using a 5% level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

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.

Cars arrive at a motorway toll booth at an average rate of 150 per hour.

Suggest a suitable distribution to model the number of cars arriving at the toll booth, X, per minute.

State clearly any assumptions you have made by suggesting this model.

Using your model,

find the probability that in any given minute

no cars arrive,

more than 3 cars arrive.

In any given 4 minute period, find m such that P(X m) = 0.0487

Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.

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 has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company’s website at their first attempt.

Explain why the Poisson distribution may be a suitable model in this case.

Find the probability that, in a randomly chosen 2 hour period,

all users connect at their first attempt,

at least 4 users fail to connect at their first attempt.

The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60.

Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5% level of significance and state your hypotheses clearly.

A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.

Find the probability that it will work continuously for 5 hours without a breakdown.

Find the probability that, in an 8 hour period,

the robot will break down at least once,

there are exactly 2 breakdowns.

In a particular 8 hour period, the robot broke down twice.

Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.

A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes. Find the probability that

fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am.

The café serves breakfast every day between 8 am and 12 noon.

Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.

A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.

Find the probability of exactly 4 faults in a 15 metre length of cloth.

Find the probability of more than 10 faults in 60 metres of cloth.

A retailer buys a large amount of this cloth and sells it in pieces of length x metres. He chooses x so that the probability of no faults in a piece is 0.80

Write down an equation for x and show that x = 1.7 to 2 significant figures.

The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.

Find the retailer’s expected profit.

A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be

more than 2 daisies,

either 5 or 6 daisies.

The botanist decides to count the number of daisies, x, in each of 80 randomly selected squares within the field. The results are summarised below

Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places.

Explain how the answers from part (c) support the choice of a Poisson distribution as a model.

Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square.

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.