5.02k Calculate Poisson probabilities

On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.

1. Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
2. Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday. The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
   She records the number of calls received in each of these 30-minute periods.
3. Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
4. Explain why the Poisson approximation may be reasonable in this case. The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
5. Write down the hypotheses for this test. The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
6. Carry out the test using a \(5 \%\) level of significance.

During the summer, mountain rescue team \(A\) receives calls for help randomly with a rate of 0.4 per day.

1. Find the probability that during the summer, mountain rescue team \(A\) receives at least 19 calls for help in 28 randomly selected days. The leader of mountain rescue team \(A\) randomly selects 250 summer days from the last few years.
   She records the number of calls for help received on each of these days.
2. Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team \(A\). Mountain rescue team \(A\) believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8
3. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly. During the summer, mountain rescue team \(B\) receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team \(A\). The random variable \(C\) is the total number of calls for help received by mountain rescue teams \(A\) and \(B\) during a period of \(n\) days in the summer.
   On a Monday in the summer, mountain rescue teams \(A\) and \(B\) each receive a call for help. Given that over the next \(n\) days \(\mathrm { P } ( C = 0 ) < 0.001\)
4. calculate the minimum value of \(n\)
5. Write down an assumption that needs to be made for the model to be appropriate.

Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.

1. Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed.
   Use a \(5 \%\) level of significance and state your hypotheses clearly. Only 1.2\% of the calls to the call centre last longer than 8 minutes.
   One day Tiang has 70 calls.
2. Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes. The call centre records show that \(95 \%\) of days have at least one call lasting longer than 30 minutes.
   On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.
3. Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.

1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

1. Find the probability that
   1. there are exactly 3 errors,
   2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
3. Find P (Type I error) for the test in part (b)

1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.

\section*{Q uestion 1 continued}

3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.

1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.

1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   1. \(\mathrm { P } ( X + Y ) > 40\),
   2. \(\operatorname { Var } ( 2 X - Y )\).
   3. State a necessary assumption for your calculations in part (i) to be valid.

2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).

1. It is given that \(W\) has a Poisson distribution.
   1. Write down the standard deviation of \(W\).
   2. Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
   3. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).

4 Leyla investigates the number of shoppers who visit a shop between 10.30 am and 11 am on Saturday mornings. She makes the following assumptions.

• Shoppers visit the shop independently of one another.
• The average rate at which shoppers visit the shop between these times is constant.
  1. State an appropriate distribution with which Leyla could model the number of shoppers who visit the shop between these times.

Leyla uses this distribution, with mean 14, as her model.

Calculate the probability that, between 10.35 am and 10.50 am on a randomly chosen Saturday, at least 10 shoppers visit the shop. Leyla chooses 25 Saturdays at random.

Find the expected number of Saturdays, out of 25, on which there are no visitors to the shop between 10.35 am and 10.50 am .

In fact on 5 of these Saturdays there were no visitors to the shop between 10.35 am and 10.50 am . Use this fact to comment briefly on the validity of the model that Leyla has used.

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

1. Calculate:
   1. \(\mathrm { P } ( X \leqslant 3 )\);
   2. \(\quad \mathrm { P } ( Y = 5 )\).
2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
   1. Write down the distribution of \(T\), stating any assumption that you make.
   2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
   3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
   4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.

1. Use a Poisson distribution to find the probability that, at a given moment,
   1. in a randomly chosen area of 3 acres there are at least 4 foxes,
   2. in a randomly chosen area of 1 acre there are exactly 2 foxes.
   3. Explain briefly why a Poisson distribution might not be a suitable model.

4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by \(R\).

1. Use an appropriate approximation to find \(\mathrm { P } ( R \geqslant 2 )\).
2. Find the smallest value of \(r\) for which \(\mathrm { P } ( R \geqslant r ) < 0.01\).

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
   1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
   2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

7 In a town, the total number, \(R\), of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

1. Using the \(10 \%\) level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of \(R\).
2. Determine, for your test in part (a), the critical region for \(R\).
3. Assuming that the offer for sale of houses on the new housing development has reduced the mean value of \(R\) to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
   (4 marks)

2 The discrete random variable \(Y\) has a Poisson distribution with mean 3 Find the value of \(\mathrm { P } ( Y > 1 )\) to three significant figures.
Circle your answer. \(0.149 \quad 0.199 \quad 0.801 \quad 0.950\)

6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6

1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
   6
3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
   6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.

7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7

1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
   7
2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
   Fully justify your answer.
   7
3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
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8 The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes. 8

1. Find the probability that:
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   1. 1. the office receives exactly 2 calls in 10 minutes; 8
      2. the office receives more than 30 calls in an hour.
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      3. The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods. Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.
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      4. The office has just received a call.
         
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   2. 1. Find the probability that the next call is received more than 10 minutes later.
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   3. (ii) Mahah arrives at the office 5 minutes after the last call was received.
      State the probability that the next call received by the office is received more than 10 minutes later. Explain your answer. \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-14_2492_1721_217_150} Additional page, if required.
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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.

1. State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second. The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
2. Find the probability that the detector detects
   (A) no neutrons in a randomly chosen second,
   (B) at least 60 neutrons in a randomly chosen period of 1 minute. A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons. If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
3. Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
4. Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.

1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\mathrm { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).