5.02i Poisson distribution: random events model

Luis makes and sells rugs. He knows that faults occur randomly in his rugs at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\)

1. Find the probability of there being exactly 5 faults in one of his rugs that is \(4 \mathrm {~m} ^ { 2 }\) in size.
2. Find the probability that there are more than 5 faults in one of his rugs that is \(6 \mathrm {~m} ^ { 2 }\) in size. Luis makes a rug that is \(4 \mathrm {~m} ^ { 2 }\) in size and finds it has exactly 5 faults in it.
3. Write down the probability that the next rug that Luis makes, which is \(4 \mathrm {~m} ^ { 2 }\) in size, will have exactly 5 faults. Give a reason for your answer. A small rug has dimensions 80 cm by 150 cm . Faults still occur randomly at a rate of 3 every \(4 \mathrm {~m} ^ { 2 }\) Luis makes a profit of \(\pounds 80\) on each small rug he sells that contains no faults but a profit of \(\pounds 60\) on any small rug he sells that contains faults. Luis sells \(n\) small rugs and expects to make a profit of at least \(\pounds 4000\)
4. Calculate the minimum value of \(n\) Luis wishes to increase the productivity of his business and employs Rhiannon. Faults also occur randomly in Rhiannon's rugs and independently to faults made by Luis. Luis randomly selects 10 small rugs made by Rhiannon and finds 13 faults.
5. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the suggestion that the rate at which faults occur is higher for Rhiannon than for Luis. State your hypotheses clearly.

1. The number of particles per millilitre in a solution is modelled by a Poisson distribution with mean 0.15

A randomly selected 50 millilitre sample of the solution is taken.

1. Find the probability that
   1. exactly 10 particles are found,
   2. between 6 and 11 particles (inclusive) are found. Petra takes 12 independent samples of \(m\) millilitres of the solution.
      The probability that at least 2 of these samples contain no particles is 0.1184
2. Using the Statistical Tables provided, find the value of \(m\)

A bakery sells muffins individually at an average rate of 8 muffins per hour.

1. Find the probability that, in a randomly selected one-hour period, the bakery sells at least 4 but not more than 8 muffins. A sample of 5 non-overlapping half-hour periods is selected at random.
2. Find the probability that the bakery sells fewer than 3 muffins in exactly 2 of these periods. Given that 4 muffins were sold in a one-hour period,
3. find the probability that more muffins were sold in the first 15 minutes than in the last 45 minutes.

1 A garage sells tyres. The number of customers arriving at the garage to buy tyres in a 10-minute period is modelled by a Poisson distribution with mean 2

1. Find the probability that
   1. fewer than 4 customers arrive to buy tyres in the next 10 minutes,
   2. more than 5 customers arrive to buy tyres in the next 10 minutes. The manager randomly selects 20 non-overlapping, 30-minute periods.
2. Find the probability that there are between 4 and 7 (inclusive) customers arriving to buy tyres in exactly 15 of these 30-minute periods. The manager believes that placing an advert in the local paper will lead to a significant increase in the number of customers arriving at the garage.
   A week after the advert is placed, the manager randomly selects a 25 -minute period and finds that 10 customers arrive at the garage to buy tyres.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the manager's belief.
   State your hypotheses clearly.
4. Explain why the Poisson distribution is unlikely to be valid for the number of tyres sold during a 10-minute period.

5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.

1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.

According to an electric company, power failures occur randomly at a rate of \(\lambda\) every 10 weeks, \(1 < \lambda < 10\)

1. Write down an expression in terms of \(\lambda\) for the probability that there are fewer than 2 power failures in a randomly selected 10 week period.
2. Write down an expression in terms of \(\lambda\) for the probability that there is exactly 1 power failure in a randomly selected 5 week period. Over a 100 week period, the probability, using a normal approximation, that fewer than 15 power failures occur is 0.0179 (to 3 significant figures).
3. 1. Justify the use of a normal approximation.
   2. Find the value of \(\lambda\). Show each stage of your working clearly.
      

3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8 The shop has 9 Birdscope cameras in stock at the start of each week. A week is selected at random.

1. Find the probability that the demand for Birdscope cameras cannot be met in this particular week. In a year, there are 50 weeks in which Birdscope cameras can be sold.
2. Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
3. Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2 The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold. Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
4. use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.

Each day a restaurant opens between 11 am and 11 pm . During its opening hours, the restaurant receives calls for reservations at an average rate of 6 per hour.

1. Find the probability that the restaurant receives exactly 1 call for a reservation between 6 pm and 7 pm . The restaurant distributes leaflets to local residents to try and increase the number of calls for reservations. After distributing the leaflets, it records the number of calls for reservations it receives over a 90 minute period. Given that it receives 14 calls for reservations during the 90 minute period,
2. test, at the \(5 \%\) level of significance, whether the rate of calls for reservations has increased from 6 per hour. State your hypotheses clearly.

7. Members of a conservation group record the number of sightings of a rare animal. The number of sightings follows a Poisson distribution with a rate of 1 every 2 months.

1. Find the smallest value of \(n\) such that the probability that there are at least \(n\) sightings in 2 months is less than 0.05
2. Find the smallest number of months, \(m\), such that the probability of no sightings in \(m\) months is less than 0.05
3. Find the probability that there is at least 1 sighting per month in each of 3 consecutive months.
4. Find the probability that the number of sightings in an 8 month period is equal to the expected number of sightings for that period.
5. Given that there were 4 sightings in a 4 month period, find the probability that there were more sightings in the last 2 months than in the first 2 months.

4. In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)

1. Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per \(\mathrm { m } ^ { 2 }\) of the peat bog to follow a Poisson distribution.
   (1) Given that the number of Common Spotted-orchids in \(1 \mathrm {~m} ^ { 2 }\) of the peat bog can be modelled by a Poisson distribution,
2. find the probability that in a randomly selected \(1 \mathrm {~m} ^ { 2 }\) of the peat bog
   1. there are exactly 6 Common Spotted-orchids,
   2. there are fewer than 10 but more than 4 Common Spotted-orchids.
      (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected \(2 \mathrm {~m} ^ { 2 }\) of the peat bog contains 11 Common Spotted-orchids.
3. Using a \(5 \%\) significance level assess Juan's belief. State your hypotheses clearly. Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
4. use a normal approximation to find the probability that in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog there are fewer than 70 Common Spotted-orchids. Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog is 0.012 A random sample of 200 non-overlapping \(20 \mathrm {~m} ^ { 2 }\) areas of the peat bog is taken.
5. Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-15_2255_50_314_34}

The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6

1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
2. Find the smallest value of \(n\) such that the probability that at least \(n\) cars enter the safari park in 10 minutes is less than 0.05 The probability that no cars enter the safari park in \(m\) minutes, where \(m\) is an integer, is less than 0.05
3. Find the smallest value of \(m\) A car enters the safari park.
4. Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
5. find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
6. Determine the critical region for the test at the \(5 \%\) level of significance.

Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.

1. Find the probability that in 15 metres of her cloth there are
   1. less than 3 faults,
   2. at least 6 faults. Each roll contains 100 metres of Bhavna's cloth.
      She selects 15 rolls at random.
2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
   Each piece of her cloth is 4 metres long.
   The cost to make each piece is \(\pounds 5.00\) She sells each piece of her cloth that contains no faults for \(\pounds 7.40\) She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.

A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts. A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$

1. show that \(\sigma = 6\)
2. Hence find the value of \(n\) and the value of \(p\)

The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8

1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera. A car has been caught speeding by this camera.
2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined £60
4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)

4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.

1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.

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A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.

1. Explain why the Poisson distribution may be a suitable model in this case. Find the probability that, in a randomly chosen \(\mathbf { 2 }\) hour period,
2. 1. all users connect at their first attempt,
   2. at least 4 users fail to connect at their first attempt. The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60 .
3. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.
   

2. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the new salesman has increased house sales.

5. An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour.
2. Find the probability that in a randomly chosen hour
   1. all Internet users connect at their first attempt,
   2. more than 4 users fail to connect at their first attempt.
3. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period.
4. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period.

2. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test \(\mathrm { H } _ { 0 } : \lambda = 7\) against \(\mathrm { H } _ { 1 } : \lambda \neq 7\).

1. Using a \(5 \%\) significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to \(2.5 \%\).
2. Write down the significance level of this test. The actual value of \(x\) obtained was 5 .
3. State a conclusion that can be drawn based on this value.

3. A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. Assuming this model and a mean of 0.7 weeds per \(\mathrm { m } ^ { 2 }\), find
2. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
3. Using a suitable approximation, find the probability that in a plot of \(100 \mathrm {~m} ^ { 2 }\) there will be more than 66 of these weeds.
   (6)

2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.

1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
2. there will be 2 accidents in the same week,
3. there is at least one accident per week for 3 consecutive weeks,
4. there are more than 4 accidents in a 2 week period.

3. (a) State two conditions under which a Poisson distribution is a suitable model to use in statistical work. The number of cars passing an observation point in a 10 minute interval is modelled by a Poisson distribution with mean 1.
(b) Find the probability that in a randomly chosen 60 minute period there will be

1. exactly 4 cars passing the observation point,
2. at least 5 cars passing the observation point. The number of other vehicles, other than cars, passing the observation point in a 60 minute interval is modelled by a Poisson distribution with mean 12.
   (c) Find the probability that exactly 1 vehicle, of any type, passes the observation point in a 10 minute period.

A robot is programmed to build cars on a production line. The robot breaks down at random at a rate of once every 20 hours.

1. Find the probability that it will work continuously for 5 hours without a breakdown. Find the probability that, in an 8 hour period,
2. the robot will break down at least once,
3. there are exactly 2 breakdowns. In a particular 8 hour period, the robot broke down twice.
4. Write down the probability that the robot will break down in the following 8 hour period. Give a reason for your answer.