5.02k Calculate Poisson probabilities

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. [1] Find the probability of
2. exactly 2 accidents in the next 6 month period, [3]
3. no accidents in exactly 2 of the next 4 months. [3]

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]
2. more than 5. [2]

Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.

1. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. [3]

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.

1. 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. [5]

From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m.

1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. [2]

The twine is usually sold in balls of length 100 m. A customer buys three balls of twine.

1. Find the probability that only one of them will have fewer than 6 faults. [6]

As a special order a ball of twine containing 500 m is produced.

1. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive. [6]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable Y ~ Po(30).

1. Estimate P(Y > 28). [6]

The random variable R has the binomial distribution B(12, 0.35).

1. Find P(R ≥ 4). [2]

The random variable S has the Poisson distribution with mean 2.71.

1. Find P(S ≤ 1). [3]

The random variable T has the normal distribution N(2.5, 5²).

1. Find P(T ≤ 18). [2]

The random variable \(R\) has the binomial distribution B(12, 0.35).

1. Find P(\(R \geq 4\)). [2]

The random variable \(S\) has the Poisson distribution with mean 2.71.

1. Find P(\(S \leq 1\)). [3]

The random variable \(T\) has the normal distribution N(25, \(5^2\)).

1. Find P(\(T \leq 18\)). [2]

1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]

A researcher has suggested that 1 in 150 people is likely to catch a particular virus. Assuming that a person catching the virus is independent of any other person catching it,

1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]

Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.

1. Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]

Find the probability that in any randomly selected 10 minute interval

1. exactly 6 cars pass this point, [3]
2. at least 9 cars pass this point. [2]

After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.

1. Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]

From past records a manufacturer of ceramic plant pots knows that 20\% of them will have defects. To monitor the production process, a random sample of 25 pots is checked each day and the number of pots with defects is recorded.

1. Find the critical regions for a two-tailed test of the hypothesis that the probability that a plant pot has defects is 0.20. The probability of rejection in either tail should be as close as possible to 2.5\%. [5]
2. Write down the significance level of the above test. [1]

A garden centre sells these plant pots at a rate of 10 per week. In an attempt to increase sales, the price was reduced over a six-week period. During this period a total of 74 pots was sold.

1. Using a 5\% level of significance, test whether or not there is evidence that the rate of sales per week has increased during this six-week period. [7]

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]

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

1. Find the probability that the box contains exactly one defective component. [2]
2. Find the probability that there are at least 2 defective components in the box. [3]
3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

1. 1. Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
   2. State the minimum number of visits required to obtain a significant result.
   [7]
2. State an assumption that has been made about the visits to the server. [1]

In a random two minute period on a Saturday the web server is visited 20 times.

1. Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]

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]

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]