5.02k Calculate Poisson probabilities

A company claims that it receives emails at a mean rate of 2 every 5 minutes.

1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
3. Find the actual level of significance of this test. To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
4. Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
5. Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men's Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.
   

1. A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake. The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains
3. 1. exactly 2 cherries,
   2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
4. Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places. Twelve packets of Rays fruit cakes are selected at random.
5. Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

A company receives telephone calls at random at a mean rate of 2.5 per hour.

1. Find the probability that the company receives
   1. at least 4 telephone calls in the next hour,
   2. exactly 3 telephone calls in the next 15 minutes.
2. Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2 The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10
3. Test at the 5\% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.
   

In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9

1. Find
   1. \(\mathrm { P } ( X > 5 )\)
   2. \(\mathrm { P } ( 4 \leqslant X < 10 )\) The length of a working day is 7 hours.
2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that

1. at least 4 customers arrive in the next 10 minutes,
2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END

1. The number of emails per hour received by a helpdesk were recorded. The results for a random sample of 80 one-hour periods are shown in the table.

1. Show that the mean number of emails per hour in the sample is 3 The manager believes that the number of emails per hour received could be modelled by a Poisson distribution. The following table shows some of the expected frequencies.

   Number of emails per hour	Expected Frequencies
   0	\(r\)
   1	11.949
   2	17.923
   3	17.923
   4	13.443
   5	\(s\)
   \(\geqslant 6\)	\(t\)

2. Find the values of \(r , s\) and \(t\), giving your answers to 3 decimal places.
3. Using a 10\% significance level, test whether or not a Poisson model is reasonable. You should clearly state your hypotheses, test statistic and the critical value used.

1. The number of jobs sent to a printer per hour in a small office is recorded for 120 hours. The results are summarised in the following table.

1. Show that the mean number of jobs sent to the printer per hour for these data is 1.75 The office manager believes that the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. The office manager uses the mean given in part (a) to calculate the expected frequencies for this model. Some of the results are given in the following table.

   Number of jobs	0	1	2	3	4	5 or more
   Expected frequency	20.85	36.49	31.93	\(r\)	\(s\)	3.95

2. Show that the value of \(s\) is 8.15 to 2 decimal places.
3. Find the value of \(r\) to 2 decimal places. The value of \(\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) for the first four frequencies in the table is 1.43
4. Test, at the \(5 \%\) level of significance, whether or not the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. Show your working clearly, stating your hypotheses, test statistic and critical value.

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.

   \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8 or more
   Expected frequency	2.42	8.46	14.80	\(r\)	15.10	10.57	6.17	3.08	\(s\)

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .

1. 1. Calculate \(\mathrm { P } ( X = 2 )\).
   2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
2. 1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
   2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

2 The number of computers, \(A\), bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, \(B\), bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .

1. 1. Calculate \(\mathrm { P } ( A = 4 )\).
   2. Determine \(\mathrm { P } ( B \leqslant 6 )\).
   3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
2. Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
3. The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
   1. Calculate the mean and variance of these data.
   2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.

2 A new information technology centre is advertising places on its one-week residential computer courses.

1. The number of places, \(X\), booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
   1. State the standard deviation of \(X\).
   2. Calculate \(\mathrm { P } ( 6 < X < 12 )\).
2. The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
   1. Write down the distribution for \(T\), the total number of places booked each week on the publishing and web design courses.
   2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
3. The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution \(\mathrm { Po } ( 12.0 )\) could provide a suitable model for the number of places booked each week on the database course.

5

1. In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random. Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
2. 1. In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people. A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
   2. In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people. A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
   3. The random variable \(T\) denotes the total number of people in the two selected towns who have this blood disorder. Write down the distribution of \(T\) and hence determine \(\mathrm { P } ( T > 16 )\).

3 Lucy is the captain of her school's cricket team.
The number of catches, \(X\), taken by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.6 . The number of run-outs, \(Y\), effected by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.15 .

1. Find:
   1. \(\mathrm { P } ( X \leqslant 1 )\);
   2. \(\mathrm { P } ( X \leqslant 1\) and \(Y \geqslant 1 )\).
2. State the assumption that you made in answering part (a)(ii).
3. During a particular season, Lucy plays in 16 cricket matches.
   1. Calculate the probability that the number of catches taken by Lucy during this season is exactly 10 .
   2. Determine the probability that the total number of catches taken and run-outs effected by Lucy during this season is at least 15 .

4

1. A discrete random variable \(X\) has a probability function defined by $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } \quad \text { for } x = 0,1,2,3,4 , \ldots \ldots$$
   1. State the name of the distribution of \(X\).
   2. Write down, in terms of \(\lambda\), expressions for \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
   3. Write down an expression for \(\mathrm { P } ( X = x + 1 )\), and hence show that $$\mathrm { P } ( X = x + 1 ) = \frac { \lambda } { x + 1 } \mathrm { P } ( X = x )$$
2. The number of cars and the number of coaches passing a certain road junction may be modelled by independent Poisson distributions.
   1. On a winter morning, an average of 500 cars per hour and an average of 10 coaches per hour pass this junction. Determine the probability that a total of at least 10 such vehicles pass this junction during a particular 1 -minute interval on a winter morning.
   2. On a summer morning, an average of 836 cars per hour and an average of 22 coaches per hour pass this junction. Calculate the probability that a total of at most 3 such vehicles pass this junction during a particular 1 -minute interval on a summer morning. Give your answer to two significant figures.
      (3 marks)

3 A large office block is busy during the five weekdays, Monday to Friday, and less busy during the two weekend days, Saturday and Sunday. The block is illuminated by fluorescent light tubes which frequently fail and must be replaced with new tubes by John, the caretaker. The number of fluorescent tubes that fail on a particular weekday can be modelled by a Poisson distribution with mean 1.5. The number of fluorescent tubes that fail on a particular weekend day can be modelled by a Poisson distribution with mean 0.5 .

1. Find the probability that:
   1. on one particular Monday, exactly 3 fluorescent light tubes fail;
   2. during the two days of a weekend, more than 1 fluorescent light tube fails;
   3. during a complete seven-day week, fewer than 10 fluorescent light tubes fail.
2. John keeps a supply of new fluorescent light tubes. More new tubes are delivered every Monday morning to replace those that he has used during the previous week. John wants the probability that he runs out of new tubes before the next Monday morning to be less than 1 per cent. Find the minimum number of new tubes that he should have available on a Monday morning.
3. Give a reason why a Poisson distribution with mean 0.375 is unlikely to provide a satisfactory model for the number of fluorescent light tubes that fail between 1 am and 7 am on a weekday.

1 The number of cars, \(X\), passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.

1. Calculate \(\mathrm { P } ( X = 2 )\).
2. 1. Write down the distribution of \(Y\), the number of cars passing along this road in a 5-minute interval.
   2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.

1 The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

1. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
2. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
   1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
   2. What assumption did you make in answering part (b)(i)?

2

1. The number of telephone calls, \(X\), received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine \(\mathrm { P } ( X = 8 )\).
2. The number of telephone calls, \(Y\), received per hour for Dr Bracken may be modelled by a Poisson distribution with mean \(\lambda\) and standard deviation 3 .
   1. Write down the value of \(\lambda\).
   2. Determine \(\mathrm { P } ( Y > \lambda )\).
3. 1. Assuming that \(X\) and \(Y\) are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
   2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
   3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.

1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.

1. 1. Write down the distribution of \(X\), the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
   2. Determine \(\mathrm { P } ( X = 20 )\).
   3. Determine \(\mathrm { P } ( 6 \leqslant X \leqslant 18 )\).
2. Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
3. When \(n\) cars pass the speed camera, the number of cars, \(Y\), that exceed 60 mph may be modelled by the distribution \(\mathrm { B } ( n , 0.2 )\). Given that \(n = 20\), determine \(\mathrm { P } ( Y \geqslant 5 )\).
4. Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .

5

1. The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
   1. at least 9 minor accidents;
   2. more than 5 but fewer than 10 minor accidents.
2. The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
3. The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
   1. state the value of \(\lambda\);
   2. determine \(\mathrm { P } ( T > 16 )\);
   3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.

1. Find the probability that the satellite detects:
   1. exactly 4 GRBs during one particular week;
   2. at least 2 GRBs on one particular day;
   3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
2. Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
   (1 mark)

5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, \(X\), within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of \(X\) can be modelled by a Poisson distribution with \(\lambda = 3.2\).

1. Use the summarised data to support Peter's belief.
2. Using Peter's model, calculate the probability that:
   1. a single placing of the quadrat contains more than 5 quartz pebbles;
   2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
   3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
3. Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
   [0pt] [3 marks]

1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:

1. a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
2. a 20 m length of tideline along this beach contains exactly 2 drinks cans;
3. a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
   [0pt] [4 marks]

2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.

1. Find the probability that he sells no copies in a particular week.
2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.

7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,

1. find the probability that the chosen patch contains
   1. no daisies,
   2. one daisy. Ten such patches are chosen. Using your answers to part (a),
2. find the probability that the total number of daisies is less than two.
3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
4. Compare your answers to parts (b) and (c).
5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.