Explain or apply conditions in context

4 The number, \(X\), of books received at a charity shop has a constant mean of 5.1 per day.

1. State, in context, one condition for \(X\) to be modelled by a Poisson distribution.
   Assume now that \(X\) can be modelled by a Poisson distribution.
2. Find the probability that exactly 10 books are received in a 3-day period.
3. Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
   The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
4. Find the probability that the total number of books and DVDs that are received at the shop in 1 day is more than 3 .

5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.

1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
2. Find the probability that exactly 8 people call in any half hour.
3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.

6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.

1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).

8 In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.

1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. Assume now that a Poisson model is appropriate.
2. Find the probability that in 10 ml there are at least 5 bacteria.
3. Find the probability that in 3.7 ml there are exactly 2 bacteria.
4. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation.

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below.

   \(x\)	0	1	2	3	4	5	\(> 5\)
   Frequency, \(f\)	18	39	20	12	8	3	0

2. Show that the sample mean is 1.62 and calculate the sample variance.
3. Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of \(X\) ? Explain your answer. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 1.62 .
4. Find \(\mathrm { P } ( X = 2 )\). Comment on your answer in relation to the data in the table.
5. Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
6. Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.

1. Explain what the statement "failures occur randomly" means.
2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).

2 A class investigated the number of dead rabbits found along a particular stretch of road.

1. The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.
2. State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means. Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.
3. Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.

6 The number of randomly occurring events in a given time interval is denoted by \(R\). In order that \(R\) is well modelled by a Poisson distribution, it is necessary that events occur independently.

1. Let \(R\) represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let \(D\) represent the number of tables booked at the restaurant on a randomly chosen day. Assume that \(D\) can be well modelled by distribution \(\operatorname { Po } ( 7 )\).
2. Find \(\mathrm { P } ( D < 5 )\).
3. Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .

8

1. A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
   1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
   2. State a different condition that is needed. Explain whether it is likely to hold in this context.
2. The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
   1. In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
   2. In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
   3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager \(A\) says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

1. Give an improved version of Manager \(A\) 's statement, and explain why the improvement is necessary.
2. Explain whether you think your improved statement is likely to hold in this context. Assume now that the number \(B\) of breakdowns per day can be modelled by the distribution \(\operatorname { Po } ( \lambda )\).
3. Given that \(\lambda = 9.0\) and \(\mathrm { P } \left( B > B _ { 0 } \right) < 0.1\), use tables to find the smallest possible value of \(B _ { 0 }\), and state the corresponding value of \(\mathrm { P } \left( B > B _ { 0 } \right)\).
4. Given that \(\mathrm { P } ( B = 2 ) = 0.0072\), show that \(\lambda\) satisfies an equation of the form \(\lambda = 0.12 \mathrm { e } ^ { k \lambda }\), for a value of \(k\) to be stated. Evaluate the expression \(0.12 \mathrm { e } ^ { k \lambda }\) for \(\lambda = 8.5\) and \(\lambda = 8.6\), giving your answers correct to 4 decimal places. What can be deduced about a possible value of \(\lambda\) ?

2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.

1. Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
2. Find the probability of at most 1 car arriving in a period of 1 minute.
3. Find the probability of more than 5 cars arriving in a period of 10 minutes.
4. State the exact distribution of the number of cars arriving in a period of 1 hour.
5. Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520.
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.

6 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\) ? Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
2. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( x \geqslant 8 )\). The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution \(\operatorname { Po } ( 7.2 )\).
3. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
4. State an assumption needed for your answer to part (c) to be valid.
5. Give a reason why the assumption in part (d) may not be valid in practice.

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.

6. Cars arrive at a motorway toll booth at an average rate of 150 per hour.

1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute.
2. State clearly any assumptions you have made by suggesting this model. Using your model,
3. find the probability that in any given minute
   1. no cars arrive,
   2. more than 3 cars arrive.
4. In any given 4 minute period, find \(m\) such that \(\mathrm { P } ( X > m ) = 0.0487\)
5. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
   January 2011

4. A website receives hits at a rate of 300 per hour.

1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
2. State two reasons for your answer to part (a). Find the probability of
3. 10 hits in a given minute,
4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.

6. Minor defects occur in a particular make of carpet at a mean rate of 0.05 per \(\mathrm { m } ^ { 2 }\).

1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer. A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires \(30 \mathrm {~m} ^ { 2 }\) of this carpet. Find the probability that the foyer carpet contains
2. exactly 2 defects,
3. more than 5 defects. The carpet fitter orders a total of \(355 \mathrm {~m} ^ { 2 }\) of the carpet for the whole hotel.
4. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects.
   (6)

3. An estate agent sells properties at a mean rate of 7 per week.

1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model.
2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties.
3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties.

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 .

3. An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution.
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players.
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days.
   (3 marks)

4 It is known that in an electronic circuit, the number of electrons passing per nanosecond can be modelled by a Poisson distribution. In a particular electronic circuit, the mean number of electrons passing per nanosecond is 12 .

1. 1. Determine the probability that there are more than 15 electrons passing in a randomly selected nanosecond.
   2. Determine the probability that there are fewer than 50 electrons passing in a randomly selected period of 5 nanoseconds.
2. Explain what you can deduce about the electrons passing in the circuit from the fact that a Poisson distribution is a suitable model.

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.