5.02i Poisson distribution: random events model

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

Find the probability that

1. 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.
2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
   

4. A website receives hits at a rate of 300 per hour.

1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
2. State two reasons for your answer to part (a). Find the probability of
3. 10 hits in a given minute,
4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.

7. (a) Explain briefly what you understand by

1. a critical region of a test statistic,
2. the level of significance of a hypothesis test.
   (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
   (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

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

1. Find the probability that in any given year there will be
   1. exactly 7 power cuts,
   2. at least 4 power cuts.
2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20

2. On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407 , to 3 significant figures. Find the probability of
2. exactly 2 accidents in the next 6 month period,
3. no accidents in exactly 2 of the next 4 months.

5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is

1. exactly 4,
2. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
4. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.

3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.

1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
2. a randomly chosen page has no misprints,
3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.

2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily's blood is found to contain only 1 deformed red blood cell. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily's blood.

An administrator makes errors in her typing randomly at a rate of 3 errors every 1000 words.

1. In a document of 2000 words find the probability that the administrator makes 4 or more errors. The administrator is given an 8000 word report to type and she is told that the report will only be accepted if there are 20 or fewer errors.
2. Use a suitable approximation to calculate the probability that the report is accepted.

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

1. Find the probability of exactly 4 faults in a 15 metre length of cloth.
2. 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
3. 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 \(\pounds 1.50\) on any pieces that do contain faults.
4. Find the retailer's expected profit.

2. 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.

1. 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.
2. 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.
3. 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.

5. 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 .

1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.

4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.

1. Find the probability that in the next 4 weeks the estate agent sells,
   1. exactly 3 houses,
   2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.

1. 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

1. exactly 4 customers join the queue in the next 10 minutes,
2. 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.
3. Find \(\mathrm { P } ( T > 3.5 )\) 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.
4. Find \(\mathrm { P } ( C \geqslant 3 )\) Bethan has just reached the front of the queue and starts paying the assistant.
5. Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.
   

6. Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.

1. Find the critical region for a two-tailed test using a \(10 \%\) level of significance. The probability of rejection in each tail should be less than 0.05
2. Find the actual significance level of this test. The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.
3. Use a suitable approximation to test the bakery's claim. You should state your hypotheses clearly and use a \(5 \%\) level of significance.

1. The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25

Find the probability that

1. a randomly chosen metre of cloth has 1 defect,
2. 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.
3. 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.

1. 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.
2. 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.
3. 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.
   

3. 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.

1. Write down the distribution of \(X\).
2. Find \(\mathrm { P } ( X > 7 )\).
3. Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).
4. Find the probability that there is at least one accident in exactly 4 of the next 6 months.

Sammy manufactures wallpaper. She knows that defects occur randomly in the manufacturing process at a rate of 1 every 8 metres. Once a week the machinery is cleaned and reset. Sammy then takes a random sample of 40 metres of wallpaper from the next batch produced to test if there has been any change in the rate of defects.

1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. You should choose your critical region so that the probability of rejection is less than 0.05 in each tail.
2. State the actual significance level of this test. Thomas claims that his new machine would reduce the rate of defects and invites Sammy to test it. Sammy takes a random sample of 200 metres of wallpaper produced on Thomas' machine and finds 19 defects.
3. Using a suitable approximation, test Thomas' claim. You should use a \(5 \%\) level of significance and state your hypotheses clearly.
   

Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.

1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
   1. exactly 7
   2. at least 10 A patient arrives at 11.30 a.m.
2. Find the probability that the next patient arrives before 11.45 a.m.
   

A company claims that it receives emails at a mean rate of 2 every 5 minutes.

1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
3. Find the actual level of significance of this test. To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
4. Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
5. Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men's Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.
   

1. 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.

1. Calculate the mean and the variance of these data.
2. 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
3. 1. exactly 2 cherries,
   2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
4. 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.
5. Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

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

1. Find the probability that the company receives
   1. at least 4 telephone calls in the next hour,
   2. exactly 3 telephone calls in the next 15 minutes.
2. 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
3. Test at the 5\% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.
   

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

1. Find
   1. \(\mathrm { P } ( X > 5 )\)
   2. \(\mathrm { P } ( 4 \leqslant X < 10 )\) The length of a working day is 7 hours.
2. 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.
3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.

1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
   (6 marks)