Joe Lucky plays the lottery on any given week with probability p, independently of whether he played on any other week. Each time he plays, he has a probability q of winning, again independently of everything else. During a fixed time period of n weeks, let X be the number of weeks that he played the lottery and Y be 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 PMF pY | X (y | x).
(c) Find the joint PMF pX,Y (x, y).
(d) Find the marginal PMF pY (y). Hint: One possibility is to start with the answer to part (c), but the algebra can be messy. But if you think intuitively about the procedure that generates Y , you may be able to guess the answer.
(e) Find the conditional PMF pX | Y (x | y). Do this algebraically using the preceding answers.
(f) Rederive the answer to part (e) by thinking as follows: for each one of the n-Y weeks that he did not win, the answer to part (a) should tell you something.