5.02i Poisson distribution: random events model

6 The number of randomly occurring events in a given time interval is denoted by \(R\). In order that \(R\) is well modelled by a Poisson distribution, it is necessary that events occur independently.

1. Let \(R\) represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let \(D\) represent the number of tables booked at the restaurant on a randomly chosen day. Assume that \(D\) can be well modelled by distribution \(\operatorname { Po } ( 7 )\).
2. Find \(\mathrm { P } ( D < 5 )\).
3. Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .

2 The random variable \(D\) has the distribution \(\operatorname { Po } ( 20 )\). Using an appropriate approximation, which should be justified, calculate \(\mathrm { P } ( D \geqslant 25 )\).

5 In a large region of derelict land, bricks are found scattered in the earth.

1. State two conditions needed for the number of bricks per cubic metre to be modelled by a Poisson distribution. Assume now that the number of bricks in 1 cubic metre of earth can be modelled by the distribution Po(3).
2. Find the probability that the number of bricks in 4 cubic metres of earth is between 8 and 14 inclusive.
3. Find the size of the largest volume of earth for which the probability that no bricks are found is at least 0.4.

1

1. The number of inhabitants of a village who are selected for jury service in the course of a 10-year period is a random variable with the distribution \(\operatorname { Po } ( 4.2 )\).
   1. Find the probability that in the course of a 10-year period, at least 7 inhabitants are selected for jury service.
   2. Find the probability that in 1 year, exactly 2 inhabitants are selected for jury service.
   3. Explain why the number of inhabitants of the village who contract influenza in 1 year can probably not be well modelled by a Poisson distribution.

6

1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
   1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
   2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).

7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).

1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.

8

1. A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
   1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
   2. State a different condition that is needed. Explain whether it is likely to hold in this context.
2. The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
   1. In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
   2. In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
   3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$

4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.

1. State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
2. Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
3. Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
4. Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.

6 At a tourist car park, a survey is made of the regions from which cars come.

1. It is given that \(40 \%\) of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
2. It is given that \(1 \%\) of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.

2 The number of neutrinos that pass through a certain region in one second is a random variable with the distribution \(\operatorname { Po } \left( 5 \times 10 ^ { 4 } \right)\). Use a suitable approximation to calculate the probability that the number of neutrinos passing through the region in 40 seconds is less than \(1.999 \times 10 ^ { 6 }\).

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager \(A\) says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

1. Give an improved version of Manager \(A\) 's statement, and explain why the improvement is necessary.
2. Explain whether you think your improved statement is likely to hold in this context. Assume now that the number \(B\) of breakdowns per day can be modelled by the distribution \(\operatorname { Po } ( \lambda )\).
3. Given that \(\lambda = 9.0\) and \(\mathrm { P } \left( B > B _ { 0 } \right) < 0.1\), use tables to find the smallest possible value of \(B _ { 0 }\), and state the corresponding value of \(\mathrm { P } \left( B > B _ { 0 } \right)\).
4. Given that \(\mathrm { P } ( B = 2 ) = 0.0072\), show that \(\lambda\) satisfies an equation of the form \(\lambda = 0.12 \mathrm { e } ^ { k \lambda }\), for a value of \(k\) to be stated. Evaluate the expression \(0.12 \mathrm { e } ^ { k \lambda }\) for \(\lambda = 8.5\) and \(\lambda = 8.6\), giving your answers correct to 4 decimal places. What can be deduced about a possible value of \(\lambda\) ?

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .

1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
2. A page is chosen at random. Find the probability of
   (A) exactly 1 error on this page,
   (B) at least 2 errors on this page. 10 pages are chosen at random.
3. Find the probability of exactly 10 errors in these 10 pages.
4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.

2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.

1. Find the probability that John observes
   (A) no butterflies in a minute,
   (B) at least 2 butterflies in a minute,
   (C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
2. Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
3. Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
4. Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.

2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.

1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
2. Find the probability that, during one minute, Jess sees
   (A) exactly one meteor,
   (B) at least 4 meteors.
3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).

2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.

1. State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
2. State the variance of this Poisson distribution.
3. Find the probability of
   (A) exactly 3 counts in a 5 -second period,
   (B) at least 3 counts in a 5 -second period.
4. Find the probability of exactly 40 counts in a period of 60 seconds.
5. Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
6. The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.

2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.

1. Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
2. Find the probability of at most 1 car arriving in a period of 1 minute.
3. Find the probability of more than 5 cars arriving in a period of 10 minutes.
4. State the exact distribution of the number of cars arriving in a period of 1 hour.
5. Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.

2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.

1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
2. Explain why \(X\) has, approximately, a Poisson distribution.
3. Use a Poisson approximating distribution to find
   (A) \(\mathrm { P } ( X = 1 )\),
   (B) \(\mathrm { P } ( X > 4 )\).
4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
5. Use this Normal approximating distribution to
   (A) find the probability that there are at least 90 births which have the mutation,
   ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.

2 Suppose that 3\% of the population of a large city have red hair.

1. A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample. A random sample of 60 people from the city is selected. The random variable \(X\) represents the number of people in this sample who have red hair.
2. Explain why the distribution of \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Hence find
   (A) \(\mathrm { P } ( X = 2 )\),
   (B) \(\mathrm { P } ( X > 2 )\).
4. Discuss whether or not it would be appropriate to model \(X\) using a Normal approximating distribution. A random sample of 5000 people from the city is selected.
5. State the exact distribution of the number of people with red hair in the sample.
6. Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

2 It was stated in 2012 that \(3 \%\) of \(\pounds 1\) coins were fakes. Throughout this question, you should assume that this is still the case.

1. Find the probability that, in a random selection of \(25 \pounds 1\) coins, there is exactly one fake coin. A random sample of \(250 \pounds 1\) coins is selected.
2. Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
3. Use a Poisson distribution to find the probability that, in this sample, there are
   (A) exactly 10 fake coins,
   (B) at least 10 fake coins.
4. Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins. It is known that \(0.2 \%\) of another type of coin are fakes.
5. A random sample of size \(n\) of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }\), where \(\lambda = 0.002 n\).
6. Use the approximation \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }\) for small values of \(\lambda\) to estimate the value of \(n\) for which the probability in part ( \(\mathbf { v }\) ) is equal to 0.995 .

2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
4. Use this Poisson distribution to find the probability that there are
   (A) exactly two genes that mutate,
   (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.

1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.