A university admissions officer admits students for two postgraduate courses, one in Management and one in Management with French Studies. On average there are 45 and 14 applicants per year for the two courses respectively who meet the suitability standards. For the Management course experience suggests that if he makes a offer to a suitable candidate, there is a 65% probability that the candidate will accept the place. For Management with French Studies the corresponding probability is 80%.

(a) Why is a Poisson distribution likely to be a reasonable probability distribution to describe the number of applicants per year to each of the courses?

(b) What is the probability there will be exactly 15 suitable applicants for the Management with French Studies course in a particular year? [Do not use tables and show your working.]

(c) What is the probability that there will be more than 12 students on the Management with French Studies course if 15 offers are made? Justify your answer.

(d) What is the probability that there will be less than 33 students on the Management course if 40 offers are made? [Use normal approximation and show your working.]

(e) For the Management course, the admissions officer has an income target that corresponds to admitting 26 students. The room used for teaching the course can seat 32 students. Without doing any further calculations outline the difficulties the admissions officer faces in trying to achieve his income target.