Consider a casino game that an individual (Joe) wants to play. It costs him N dollars each time to play. He loves this game and wants to continue playing until he is either broke or he breaks the bank (wins all the money). The probability of winning is p; the probability of losing is q. These are fixed probability values every time the game is played.

Joe brought $M to the casino. Every time one plays you either lose the entrance fee ($N) or you win and are paid back D dollars.

(a) How much money does Joe expect to have after playing n times? Derive a formula for how much money he has.

(b) Suppose Joe starts with $100, p=0.3, q=0.7, N=$5, and D=$20. Is it likely that Joe will break the bank?

(c) If the answer to (b) is no, how many times is it likely that Joe can play this game before he is broke?