5.02i Poisson distribution: random events model

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

1. more than 2 daisies, [3]
2. either 5 or 6 daisies. [2]

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 $$\sum x = 295 \quad \sum x^2 = 1386$$

1. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places. [3]
2. Explain how the answers from part (c) support the choice of a Poisson distribution as a model. [1]
3. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square. [2]

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]

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

1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute. [2]
2. State clearly any assumptions you have made by suggesting this model. [2]

Using your model,

1. find the probability that in any given minute
   1. no cars arrive,
   2. more than 3 cars arrive.
   [3]
2. In any given 4 minute period, find \(m\) such that P(\(X > m\)) = 0.0487 [3]
3. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period. [6]

1. Explain what you understand by a critical region of a test statistic. [2]

The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac{1}{7}\).

1. Find the probability that on a particular day there are fewer than 2 breakdowns. [3]
2. Find the probability that during a 14-day period there are at most 4 breakdowns. [3]

The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.

1. Test, at the 5% level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly. [7]

Minor defects occur in a particular make of carpet at a mean rate of 0.05 per m\(^2\).

1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer.

A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires 30 m\(^2\) of this carpet. Find the probability that the foyer carpet contains

1. exactly 2 defects, [3]
2. more than 5 defects. [3]

The carpet fitter orders a total of 355 m\(^2\) of the carpet for the whole hotel.

1. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects. [6]

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]

An insurance company conducts its business by using a Call Centre. The average number of calls per minute is 3.5. In the first minute after a TV advertisement is shown, the number of calls received is 7.

1. Stating your hypotheses carefully, and working at the 5\% significance level, test whether the advertisement has had an effect. [5 marks]
2. Find the number of calls that would be required in the first minute for the null hypothesis to be rejected at the 0.1\% significance level. [3 marks]

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]

Another machine produces biscuits with a mean of 1.9 currants per biscuit.

1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]

A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5.

1. Find P\((X = 0)\). [1 mark]
2. In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\). [8 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]

A secretarial agency carefully assesses the work of a new recruit, with the following results after 150 pages:

No of errors	0	1	2	3	4	5	6
No of pages	16	38	41	29	17	7	2

1. Find the mean and variance of the number of errors per page. [4 marks]
2. Explain how these results support the idea that the number of errors per page follows a Poisson distribution. [1 mark]
3. After two weeks at the agency, the secretary types a fresh piece of work, six pages long, which is found to contain 15 errors. The director suspects that the secretary was trying especially hard during the early period and that she is now less conscientious. Using a Poisson distribution with the mean found in part (a), test this hypothesis at the 5% significance level. [5 marks]

A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that

1. he is not late at all, [2 marks]
2. he is late more than twice. [3 marks]

In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.

1. Use the Normal approximation to estimate the probability that he loses his privileges. [7 marks]

A certain type of steel is produced in a foundry. It has flaws (small bubbles) randomly distributed, and these can be detected by X-ray analysis. On average, there are 0·1 bubbles per cm³, and the number of bubbles per cm³ has a Poisson distribution. In an ingot of 40 cm³, find

1. the probability that there are less than two bubbles, [3 marks]
2. the probability that there are more than 3 but less than 10 bubbles. [3 marks]

A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per cm³. In a sample ingot of 60 cm³, there is just one bubble.

1. Carry out a hypothesis test at the 1% significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully. [6 marks]

In a survey of 22 families, the number of children, \(X\), in each family was given by the following table, where \(f\) denotes the frequency:

\(X\)	0	1	2	3	4	5
\(f\)	3	8	5	3	2	1

1. Find the mean and variance of \(X\). [4 marks]
2. Explain why these results suggest that \(X\) may follow a Poisson distribution. [1 mark]
3. State another feature of the data that suggests a Poisson distribution. [1 mark]

It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2·4. Assuming that this is correct,

1. find the probability that a family has less than two children. [3 marks]
2. Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children. [3 marks]

A centre for receiving calls for the emergency services gets an average of 3.5 emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,

1. find the probability that more than 6 calls arrive in any particular minute. [3 marks] Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
2. Find how many operators are required for there to be a 99\% probability that a call can be dealt with immediately. [3 marks] It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis \(H_0\) that there is no disaster,
3. find the number of calls that need to be received in one minute to disprove \(H_0\) at the 0.1 \% significance level. [3 marks]

A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:

No of counts, \(X\)	0	1	2	3	4	5	6
Frequency	10	24	29	16	12	6	3

1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution. [5 marks]
2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute. [4 marks]
3. Find the number of one-minute intervals, in the sample of 100, in which more than 6 counts would be expected. [2 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]