5.02k Calculate Poisson probabilities

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h] \end{table}

1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D	E
   \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	\(\leqslant 2\)	7	0.1165	6.9887	0.0000
   3	3	5		8.0874	1.1786
   4	4	12			0.2765
   5	5	10			0.0255
   6	6	10	0.1490	8.9400	0.1257
   7	7	11	0.1086	6.5134	3.0904
   8	\(\geqslant 8\)	5	0.1440	8.6414
   9

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the hypothesis test at the 5\% significance level.
3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h] \end{table}

1. Use the results to find an estimate for \(\mu\). The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2}

   \multirow[b]{2}{*}{1}	A	B	C	D	E
   	Number of vehicles	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	36	0.2725	27.2532	2.8073
   3	1	33	0.3543	35.4291
   4	2	14			3.5400
   5	\(\geqslant 3\)	17			0.5145
   6

   \end{table}
2. Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level.
3. Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.

5 Over a long period of time, it is found that the mean number of mistakes made by a certain player when playing a particular piece of music is 5 . The number of mistakes that the player makes when playing the piece is denoted by the random variable \(Y\).

1. State two assumptions necessary for \(Y\) to be modelled by a Poisson distribution. For the remainder of this question you may assume that \(Y\) can be modelled by a Poisson distribution.
2. 1. Find the probability that the player makes exactly 3 mistakes when playing the piece.
   2. Find the probability that the player makes fewer than 3 mistakes when playing the piece.
   3. Find the probability that the player makes fewer than 6 mistakes in total when playing the piece twice, assuming that the performances are independent. In a recording studio, the player plays the piece once in the morning and once in the afternoon each day for one week (7 days). It can be assumed that all the performances are independent of each other. The performances are recorded onto two CDs, one for each of two critics, A and B, to review. The critics are interested in the total number of mistakes made by the player per day. Unfortunately, there is a recording error in one of the CDs; on this CD, every piece that is supposed to be an afternoon recording is in fact just a repeat of that morning's recording. The random variables \(M _ { 1 }\) and \(M _ { 2 }\) represent the total number of mistakes per day for the correctly recorded CD and for the wrongly recorded CD respectively.
3. By considering the values of \(\mathrm { E } \left( M _ { 1 } \right)\) and \(\mathrm { E } \left( M _ { 2 } \right)\) explain why it is not possible to use the mean number of mistakes per day on the CDs to determine which critic received the wrongly recorded CD. Each critic counts the total number of mistakes made per day, for each of the 7 days of recordings on their CD. Summary data for this is given below. Critic A: \(\quad n = 7 , \quad \sum x _ { A } = 70 , \quad \sum x _ { A } ^ { 2 } = 812\) Critic B: \(\quad \mathrm { n } = 7 , \sum \mathrm { x } _ { \mathrm { B } } = 72 , \sum \mathrm { x } _ { \mathrm { B } } ^ { 2 } = 800\)
4. By considering the values of \(\operatorname { Var } \left( M _ { 1 } \right)\) and \(\operatorname { Var } \left( M _ { 2 } \right)\) determine which critic is likely to have received the wrongly recorded CD.

2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
   For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).

5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.

1. 1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
   2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
3. Determine \(\mathrm { P } ( X < 5 )\).

2 A special railway coach detects faults in the railway track before they become dangerous.

1. Write down the conditions required for the numbers of faults in the track to be modelled by a Poisson distribution. You should now assume that these conditions do apply, and that the mean number of faults in a 5 km length of track is 1.6 .
2. Find the probability that there are at least 2 faults in a randomly chosen 5 km length of track.
3. Find the probability that there are at most 10 faults in a randomly chosen 25 km length of track.
4. On a particular day the coach is used to check 10 randomly chosen 1 km lengths of track. Find the probability that exactly 1 fault, in total, is found.

1 During a meteor shower, the number of meteors that can be seen at a particular location can be modelled by a Poisson distribution with mean 1.2 per minute.

1. Find the probability that exactly 2 meteors are seen in a period of 1 minute.
2. Find the probability that more than 3 meteors are seen in a period of 1 minute.
3. Find the probability that no more than 8 meteors are seen in a period of 10 minutes.
4. Explain what the fact that the number of meteors seen can be modelled by a Poisson distribution tells you about the occurrence of meteors.

1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).

1. Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
2. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
3. Use a Poisson distribution to calculate each of the following probabilities.
   Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.

2 The number of cars arriving per minute to queue at a drive-through fast-food restaurant is modelled by the random variable \(X\). The standard deviation of \(X\) is 0.6 . You should assume that arrivals are random and independent and occur at a constant average rate.

1. Find the mean of \(X\).
2. 1. Calculate \(\mathrm { P } ( X = 1 )\).
   2. Calculate \(\mathrm { P } ( X > 1 )\).
3. Find the probability that fewer than 5 cars arrive in a randomly chosen 20 -minute period.

3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.

1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
3. What is the probability that her father gives her the car?

3. Two basketball players, Steph and Klay, score baskets at random at a rate of \(2 \cdot 1\) and \(1 \cdot 9\) respectively per quarter of a game. Assume that baskets are scored independently, and that Steph and Klay each play all four quarters of the game.

1. Stating the model that you are using, find the probability that they will score a combined total of exactly 20 baskets in a randomly selected game.
2. A quarter of a game lasts 12 minutes.
   1. State the distribution of the time between baskets for Steph. Give the mean and standard deviation of this distribution.
   2. Given that Klay scores at the end of the third minute in a quarter of a game, find the probability that Klay doesn't score for the rest of the quarter.
3. When practising, Klay misses \(4 \%\) of the free throws he takes. One week he takes 530 free throws. Calculate the probability that he misses more than 25 free throws.

Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.

1. 1. Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
   2. Justify any distribution you have used in answering (a)(i).
      
   (b) On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
   1. find the expected time he catches his first fish,
   2. calculate the probability that he will not catch a fish by 3 pm .
      
   (c) On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
   [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
2. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
   PLEASE DO NOT WRITE ON THIS PAGE

8 The time in hours to failure of a component may be modelled by an exponential distribution with parameter \(\lambda = 0.025\) In a manufacturing process, the machine involved uses one of these components continuously until it fails. The component is then immediately replaced.
8

1. Write down the mean time to failure for a component. 8
2. Find the probability that a component will fail during a 12-hour shift. 8
3. A component has not failed for 30 hours. Find the probability that this component lasts for at least another 30 hours.
   [0pt] [2 marks] 8
4. Find the probability that a component does not fail during 4 consecutive 12-hour shifts.
   [0pt] [3 marks]
   8
5. (i) State the distribution that can be used to model the number of components that fail during one hour of the manufacturing process.
   [0pt] [2 marks]
   8 (e) (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
   [0pt] [2 marks]

The number of heaters, \(H\), bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7

1. Calculate \(\mathrm { P } ( H \geqslant 2 )\) The number of heaters, \(G\), bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where \(G\) and \(H\) are independent.
2. Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places.
3. Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days. December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14
4. Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.

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Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.

1. Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work. Andreia asks the secretary to produce a 250 -word article for a magazine.
2. Find the probability that there are exactly 5 errors in this article. Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words. In scheme \(\mathbf { A }\) the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors. In scheme \(\mathbf { B }\) the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56
3. Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose. Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test.

A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.

1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day. The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
3. 1. find the expected number of days that the company receives more than 28 call-outs,
   2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
   3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.

Rowan and Alex are both check-in assistants for the same airline. The number of passengers, \(R\), checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28

1. Calculate \(\mathrm { P } ( R \geqslant 23 )\) The number of passengers, \(A\), checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where \(R\) and \(A\) are independent. A randomly selected 30-minute period is chosen.
2. Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods.
3. Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working.

Xena catches fish at random, at a constant rate of 0.6 per hour.

1. Find the probability that Xena catches exactly 4 fish in a 5 -hour period. The probability of Xena catching no fish in a period of \(t\) hours is less than 0.16
2. Find the minimum value of \(t\), giving your answer to one decimal place. Independently of Xena, Zion catches fish at random with a mean rate of 0.8 per hour.
   Xena and Zion try using new bait to catch fish. The number of fish caught in total by Xena and Zion after using the new bait, in a randomly selected 4-hour period, is 12
3. Use a suitable test to determine, at the \(5 \%\) level of significance, whether or not there is evidence that the rate at which fish are caught has increased after using the new bait. State your hypotheses clearly and the \(p\)-value used in your test.

1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.

The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907

1. Find the value of \(A\) The tablecloths are sold in packets of 20
   A randomly selected packet is taken.
2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
   The random variable \(X\) represents the number of these tablecloths that have no faults.
3. Find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
   The piece of cloth has area 30 square metres and is found to have 6 faults.
5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
6. Write down the \(p\)-value for the test used in part (e).

1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.

\begin{table}[h] \end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h] \end{table}

1. State suitable hypotheses for a test to investigate Anja's belief.
2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
3. Showing your working clearly, complete Table 2
4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
   1. 1
   2. 3
5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.

1. A manager keeps a record of accidents in a canteen.

Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.

1. Give a reason why a Poisson distribution could be a suitable model in this situation.
2. Assuming that a Poisson model is suitable, find the probability of
   1. at least 3 accidents in the next month,
   2. no more than 10 accidents in a 3-month period,
   3. at least 2 months with no accidents in an 8-month period. One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents. Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
3. state
   The manager introduces some new procedures to try and reduce the average number of accidents per month. During the following 12 months the total number of accidents is 22 The manager claims that the accident rate has been reduced.
4. Use a \(5 \%\) level of significance to carry out a suitable test to assess the manager's claim.
   You should state your hypotheses clearly and the \(p\)-value used in your test.

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

The discrete random variable \(X\) follows a Poisson distribution with mean 1.4

1. Write down the value of
   1. \(\mathrm { P } ( \mathrm { X } = 1 )\)
   2. \(\mathrm { P } ( \mathrm { X } \leqslant 4 )\) The manager of a bank recorded the number of mortgages approved each week over a 40 week period.

      Number of mortgages approved	0	1	2	3	4	5	6
      Frequency	10	16	7	4	2	0	1

   (b) Show that the mean number of mortgages approved over the 40 week period is 1.4 The bank manager believes that the Poisson distribution may be a good model for the number of mortgages approved each week. She uses a Poisson distribution with a mean of 1.4 to calculate expected frequencies as follows.

   Number of mortgages approved	0	1	2	3	4	5 or more
   Expected frequency	9.86	r	9.67	4.51	1.58	s

2. Find the value of r and the value of s giving your answers to 2 decimal places. The bank manager will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution.
3. Calculate the test statistic and state the conclusion for this test. State clearly the degrees of freedom and the hypotheses used in the test. \section*{Q uestion 4 continued} \section*{Q uestion 4 continued}

Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.

1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20 -minute period one morning. Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5 -minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
2. Find the probability that in exactly 3 of these periods there were no calls. On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
3. Find the probability that Indre missed exactly 1 call in each of these 2 breaks.

1. The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.
Using this information,

1. 1. find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,
   2. find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning. A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10
2. Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly. A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,
3. calculate the probability of a Type II error.