Joint probability of separate processes

6 At a petrol station cars arrive independently and at random times at constant average rates of 8 cars per hour travelling east and 5 cars per hour travelling west.

1. Find the probability that, in a quarter-hour period,
   (a) one or more cars travelling east and one or more cars travelling west will arrive,
   (b) a total of 2 or more cars will arrive.
2. Find the approximate probability that, in a 12 -hour period, a total of more than 175 cars will arrive.

7 The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.

1. Find the probability that, in a half-hour period,
   (a) 2 or more men and 1 or more women will visit the clinic,
   (b) a total of 3 or more people will visit the clinic.
2. Find the probability that, in a 10 -hour period, a total of more than 60 people will visit the clinic.

7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

3 Lucy is the captain of her school's cricket team.
The number of catches, \(X\), taken by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.6 . The number of run-outs, \(Y\), effected by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.15 .

1. Find:
   1. \(\mathrm { P } ( X \leqslant 1 )\);
   2. \(\mathrm { P } ( X \leqslant 1\) and \(Y \geqslant 1 )\).
2. State the assumption that you made in answering part (a)(ii).
3. During a particular season, Lucy plays in 16 cricket matches.
   1. Calculate the probability that the number of catches taken by Lucy during this season is exactly 10 .
   2. Determine the probability that the total number of catches taken and run-outs effected by Lucy during this season is at least 15 .

1 The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

1. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
2. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
   1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
   2. What assumption did you make in answering part (b)(i)?

5

1. The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
   1. at least 9 minor accidents;
   2. more than 5 but fewer than 10 minor accidents.
2. The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
3. The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots
   0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
   1. state the value of \(\lambda\);
   2. determine \(\mathrm { P } ( T > 16 )\);
   3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

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}