A gambler starts with an initial fortune of $i>0 and then on each successive gamble either wins $1 or loses $1, independently of the past, with respective probability p and q=1-p. The gambler's goal is to acquire a fortune of $N, without first getting ruined, and win the game. Assume that the gambler stops playing after winning or being ruined, whichever comes first. Let P_i denote the probability that the gambler wins before being ruined, when starting with an initial fortune of $i>0.
(a) Find P_7 when p=0.53 and N=15.
(b) Find the probability that the gambler will become infinitely rich and will never get ruined when p=0.53.
(c) Find P_25 when p=0.42 and N=40.
(d) Find the probability that the gambler will become infinitely rich and will never get ruined when p=0.42.