19. Alberto and Angela play backgammon regularly. The probability that Alberto wins a game depends on whether he won or lost the previous game. It is p for Alberto to win a game if he lost the previous game, and p to lose a game if he won the previous one.

(a) For n > 1, show that if Alberto wins the first game, the probability is 1

2

+

1 2

(1 − 2p)n that he will win the nth game.

(b) Find the expected value of the number of games Alberto will play between two consecutive wins.