N2010 P2 Q11

In this question you should state clearly all distributions that you use, together with the values of the appropriate parameters.

The number of telephone calls received by a call centre in one minute is a random variable with the distribution \(\mathrm{Po}(3)\).

1. Find the probability that exactly 8 calls are received in a randomly chosen period of 4 minutes.[2]
2. Find the length of time, to the nearest second, for which the probability that no calls are received is 0.2.[3]
3. Use a suitable approximation to find the probability that, on a randomly chosen working day of 12 hours, more than 2200 calls are received.[4]

A working day of 12 hours on which more than 2200 calls are received is said to be ‘busy’.

4. Find the probability that, in six randomly chosen working days, exactly two are busy.[2]
5. Use a suitable approximation to find the probability that, in 30 randomly chosen working days of 12 hours, fewer than 10 are busy.[4]