61. Suppose in Exercise 29 that the shooting ends when the target has been hit twice.

Let mi denote the mean number of shots needed for the ?rst hit when player i shoots ?rst, i = 1, 2. Also, let Pi , i = 1, 2, denote the probability that the ?rst hit is by player 1, when player i shoots ?rst.

(a) Find m1 and m2.

(b) Find P1 and P2.

For the remainder of the problem, assume that player 1 shoots ?rst.

(c) Find the probability that the ?nal hit was by 1.

(d) Find the probability that both hits were by 1.

(e) Find the probability that both hits were by 2.

(f) Find the mean number of shots taken.