Joe may play the lottery on any given week with probabilityp, independently of whether he played on any other week. Each time he plays, he has a probabilityqof winning, again independently of everything else. During a fixed time period ofnweeks, letXbe the number of weeks that he played the lottery andYbe the number of weeks that he won.

(a) What is the probability that he played the lottery on any particular week, given that he did not win on that week?

b) Find the conditional PMFpY|X(y|x).

(c) Find the joint PMFpX,Y(x, y).

(d) Find the marginal PMFpY(y).

(e) Find the conditional PMFpX|Y(x|y).