QUESTION 1

(a) Mr Hail, Miss Rain and Mrs Snow all apply for a job as a park keeper. Whoever one gets the job, will have to open the gates of the park at 5.30am. If Mr Hail gets the job, he will only open the gates on time one third of the time. If Miss Rain gets the job, she will only open the gates on time half of the time. Mrs Snow is an insomniac and thus she will always open the gates punctually. Given that Mr Hail has a 50% chance of getting the job and that Miss Rain and Mrs Snow have a 30% and 20% chance respectively of getting the job

i What is the probability that the park gates will be opened on time?

ii If the gates are opened on time, what is the probability that Miss Rain got the job?

(b) Historical records of rainstorms in a town indicate that on the average there are three rainstorms per year. Assuming that the occurrence of rainstorms follows a Poisson distribution what is the probability that-

i There are no rainstorms next year

ii There are exactly three rainstorms during the next two years

iii There are at least two rainstorms next year

iv There are at least two rainstorms next year, but none the year after?

(c) Use the normal approximation to the Poisson distribution to calculate the approximate probability that in the town mentioned in part (b) above-

i There will be at least forty rainstorms during the next twelve years,

ii There will be no more than fifty-five rainstorms in the next twenty years