4. Gambler's ruin . Fix a probability p (0, 1). Play a sequence of games; in each game you (independently) win $1 with probability p and lose $1 with probability q and you make nothing with probability 1 p q. Assume that your initial capital is i dollars and that you play until you either reach a predetermined amount N, or you lose all your money. You are interested in the probability Pi that you leave the game happy with your desired amount N.

(a) Compute is p0, pN

(b) Write the recurrence relationship which expresses pn in terms of pn1, pn+1

(c) Find the probability of winning when N=100, and p = .6, q = .4 and initial money i = 30