23. Consider the gambler’s ruin problem where on each bet the gambler either wins 1 with probability p or loses 1 with probability 1 − p. The gambler will continue to play until his winnings are either N − i or −i. (That is, starting with i the gambler will quit when his fortune reaches either N or 0.) Let T denote the number of bets made before the gambler stops. Use Wald’s equation, along with the known probability that the gambler’s final winnings are N − i, to find E[T].

Hint: Let Xj be the gambler’s winnings on bet j, j  1. What are the possible values of

Tj

=1 Xj? What is E

&Tj

=1 Xj

'

?