Carlos "Coyote" Jones owns quite a few harmonicas. In particular, he has 7 professional harmonicas and 12 cheaper harmonicas. They have relatively similar shapes, so when he reaches into his harmonica container without looking, he does not notice a difference between them. Suppose that he grabs 8 harmonicas, without replacement, and all selections are equally likely.
a. How many professional harmonicas does he expect to select?
b. What is the variance of the number of professional harmonicas that he selects?
c. What is the probability that exactly 5 out of the 8 harmonicas are professional?
d. What is the probability that he selects exactly 5 professional harmonicas, given that he selects at least 5 professional harmonicas?