5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]

The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]

Medical research into the nature of the disease discovers that it can be passed from one patient to another.

1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]

A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2. [2]
2. Find the variance of the number of errors per page in this sample. [2]
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]

Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.

1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.

Number of accidents	0	1	2	3	4	5
Frequency	47	57	46	35	9	6

1. Show that the mean number of accidents per day for these data is 1.6 [1]

A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.

Number of accidents	0	1	2	3	4	5 or more
Frequency	40.38	64.61	\(r\)	27.57	11.03	\(s\)

1. Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places. [3]
2. Stating your hypotheses clearly, use a 10\% level of significance to test the motorway supervisor's belief. Show your working clearly. [7]

The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.

No of hurricanes, \(h\)	0	1	2	3	4	5	6	7
Frequency	0	2	5	17	20	12	12	12

Table 1

1. Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution. [2]
2. Show that the mean number of hurricanes per year from Table 1 is 4.4875 [2]
3. Use the answer in part (b) to calculate the expected frequencies \(r\) and \(s\) given in Table 2 below to 2 decimal places. [3]

\(h\)	0	1	2	3	4	5	6	7 or more
Expected frequency	0.90	4.04	\(r\)	13.55	\(s\)	13.65	10.21	13.39

Table 2

1. Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly. [6]

The number of telephone calls received, during an \(8\)-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of \(7\).

1. Determine the probability that, during a given \(8\)-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
   1. at most \(5\); [1 mark]
   2. exactly \(7\); [2 marks]
   3. at least \(5\) but fewer than \(10\). [3 marks]
2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
3. The IT company has \(4\) engineers available for urgent visits and it may be assumed that each of these engineers takes exactly \(1\) hour for each such visit. At \(10\)am on a particular day, all \(4\) engineers are available for urgent visits.
   1. State the maximum possible number of telephone calls received between \(10\)am and \(11\)am that request an urgent visit and for which an engineer is immediately available. [1 mark]
   2. Calculate the probability that at \(11\)am an engineer will not be immediately available to make an urgent visit. [4 marks]
4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]

The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:

1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]

A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,

1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page. [1 mark]
2. Find the probability that a particular page has more than 2 misprints. [3 marks]
3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]

Chapter 2 is longer, at 20 pages.

1. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. [7 marks]

Buttercups in a meadow are distributed independently of one another and at a constant average incidence of 3 buttercups per square metre.

1. Find the probability that in 1 square metre there are more than 7 buttercups. [2]
2. Find the probability that in 4 square metres there are either 13 or 14 buttercups. [3]
3. Use a suitable approximation to find the probability that there are no more than 69 buttercups in 20 square metres. [5]
4. 1. Without using an approximation, find an expression for the probability that in \(m\) square metres there are at least 2 buttercups. [2]
   2. It is given that the probability that there are at least 2 buttercups in \(m\) square metres is 0.9. Using your answer to part (a), show numerically that \(m\) lies between 1.29 and 1.3. [4]

In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.

1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. [2]

Assume now that a Poisson model is appropriate.

1. Find the probability that in 10 ml there are at least 5 bacteria. [2]
2. Find the probability that in 3.7 ml there are exactly 2 bacteria. [3]
3. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation. [7]

It is given that \(Y \sim\) Po\((\lambda)\), where \(\lambda \neq 0\), and that P\((Y = 4) =\) P\((Y = 5)\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of P\((Y = 5)\). [5]

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]

The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]

An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.

1. Show that the sample mean is 0.4375. [1]
2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]

The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.

1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]

The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.

1. Assuming that free repairs are required independently, find the probability that
   1. the two appliances need a total of 3 free repairs between them,
   2. each appliance needs exactly one free repair.
   [5]

An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players. [3 marks]
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]

A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.

1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed. [1 mark]
2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
3. Find the probability that on a weekday morning the shop sells
   1. more than 4 pairs in a one-hour period,
   2. no pairs in a half-hour period,
   3. more than 4 pairs during each hour from 9 am until noon. [6 marks]

The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop's sales.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of an increase in sales following the appearance of the advert. [4 marks]

An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.

1. Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
2. Explain why the parameter used with this model may need to be changed if
   1. 50 g of nuts are used instead of breadcrumbs,
   2. 100g of breadcrumbs are used.
   [2 marks]

A bird table in a rural garden has 50 g of breadcrumbs spread on it. Find the probability that

1. exactly 6 different species visit the table, [2 marks]
2. more than 2 different species visit the table. [4 marks]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives

1. 3 complaints, [3 marks]
2. 10 or more complaints. [2 marks]

The supermarket is open on six days each week.

1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]

A rugby player scores an average of 0.4 tries per match in which he plays.

1. Find the probability that he scores 2 or more tries in a match. [5 marks]

The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]

A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008

1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]

A charity receives donations of more than £10000 at an average rate of 25 per year. Find the probability that the charity receives

1. exactly 30 such donations in one year, [3]
2. less than 3 such donations in one month. [5]
3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than £10000 in the next two years. [5]

A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.

1. Suggest, with reasons, a suitable distribution for modelling this situation. [3]

The student's data is shown in the table below.

1. Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). [6]

An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning. [5]

There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day. Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time. On a randomly chosen day, he records a total of 16 births from the two hospitals.

1. Investigate Stuart's claim, using a suitable test at the 5% level of significance. [6 marks]
2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. [3 marks]