5.02k Calculate Poisson probabilities

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]

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]

1. Define
   1. a Type I error,
   2. a Type II error. [2]

A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.

1. [(b)] Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
2. Calculate the probability of the Type I error for this test. [3]
3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]

It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses H₀: \(\lambda = 8\) against H₁: \(\lambda \neq 8\), using a 10\% level of significance.

1. Find the critical region for this test, such that the probability in each tail is as close as possible to 5\%. [5]
2. Given that \(\lambda = 10\), find
   1. the probability of a type II error,
   2. the power of the test. [4]

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

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]

Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

1. State these two assumptions. [2]
2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\mathrm{Po}(0.8)\).

1. 1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
   2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\mathrm{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]

1. State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day. [2]

Jane records the number of junk emails which she receives each day. During working hours (\(9\)am to \(5\)pm, Monday to Friday) the mean number of junk emails is \(7.4\) per day. Outside working hours (\(5\)pm to \(9\)am), the mean number of junk emails is \(0.3\) per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".

1. Find the probability that the number of junk emails which she receives between \(9\)am and \(5\)pm on a Monday is
   1. exactly \(10\), [1]
   2. at least \(10\). [2]
2. 1. What assumption must you make to calculate the probability that the number of junk emails which she receives from \(9\)am Monday to \(9\)am Tuesday is at most \(20\)? [1]
   2. Find the probability. [2]