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 their winnings are either N −i or −i. (That is, starting with i the gambler will quit when their fortune reaches either N or 0.) Let T denote the number of bets made before the gambler stops. UseWald’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

T j=1Xj ? What is E

/T j=1Xj 0

?