Joint probability of independent events

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

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 .