State conditions only

3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.

1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
2. a randomly chosen page has no misprints,
3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.

4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.

1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
2. less than two visitors in a 10 -minute interval,
3. at least ten visitors in a 15-minute interval.
4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
   (5 marks)

1 Over a period of time, radioactive substances decay into other substances. During this decay a Geiger counter can be used to detect the number of radioactive particles that the substance emits. A certain radioactive source is decaying at a constant average rate of 6.1 particles per 10 seconds. The particles are emitted randomly and independently of each other.

1. State a distribution which can be used to model the number of particles emitted by the source in a 10-second period.
2. State the variance of this distribution.
3. Find the probability that at least 6 particles are detected in a period of 10 seconds.
4. Find the probability that at least 36 particles are detected in a period of 60 seconds.
5. Another radioactive source emits particles randomly and independently at a constant average rate of 1.7 particles per 5 seconds. Find the probability that at least 10 but no more than 15 particles are detected altogether from the two sources in a period of 10 seconds.

2 On a car assembly line, a robot is used for a particular task.

1. State the conditions under which a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week. It is given that the average number of breakdowns of the robot in a week is 1.7 . For the remainder of this question, you should assume that a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week.
2. 1. Find the probability that the number of breakdowns of the robot in a week is exactly 4.
   2. Determine the probability that the number of breakdowns of the robot in a week is at least 2 .
3. Determine the probability that the number of breakdowns of the robot in 52 weeks is less than 100.

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 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. 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
   • a property of the Poisson distribution that the manager should consider when deciding how to record this situation
   • whether the manager should record this as one or two accidents
   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.

4. The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.

1. State a further assumption needed for the Poisson model to be valid.
2. State the value of the standard deviation of \(R\).
3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\mathrm { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.

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.

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
2. (a) Write down an expression for \(\mathrm { P } ( X = r )\).
   (b) Hence find \(\mathrm { P } ( X = 3 )\).
3. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32 . Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.