Problem 4. Suppose that the number of calls per hour to an answering service follows a Poisson process with rate 10 per hour. Suppose that each call, independently, is made by a man with probability 2/3, and by a woman with probability 1/3, and the gender of the caller is independent of the time of the call.

(a) (4 pts) What is the probability that in one hour exactly 4 men and 2 women will call the answering service?

(b) (6 pts) What is the probability 5 men will make phone calls before 4 women do?

(c) (2 pts) Given that exactly 6 people called the answering service within the first hour, what is the probability that exactly 2 men and 2 women called in the first half hour?