Expected duration by martingales

(a) Show that $X_n - (p - q)n$ is a martingale for the walk of 3.6 and use optional stopping to derive $D_a = [N(1 - P_a) - a]/(p - q)$. Justify optional stopping. (b) Check numerically that this agrees with (3.17) for European roulette ($p = 18/37$, $a = 10$, $N = 20$). (c) A trader with a negative edge ($p = 0.48$ at even money) and capital of 20 bet units trades until ruin. How many trades does ruin take on average?