Martingale Optional Stopping Theorem

A gambler starts with $\$50$ and repeatedly bets $\$1$ on a fair coin (heads: +1, tails: -1). The gambler stops when their fortune reaches $\$0$ (ruin) or $\$100$ (target). (a) Using the optional stopping theorem, find the probability of reaching $\$100$ before $\$0$. (b) Find the expected number of bets until the game ends. (c) A new gambler plays the same game but the coin has $P(H) = 0.51$. Set up the martingale to find the ruin probability.