A random walk starts at 0 and moves up 1 with probability 2/3 and down 1 with probability 1/3 at each step. What is the probability it ever reaches -5?

A random walk starts at 0 and moves up 1 with probability 2/3 and down 1 with probability 1/3 at each step. What is the probability it ever reaches -5?

Approach: Find the constant c for which c raised to the current position is a martingale, or set up the one-step-down probability directly and solve the resulting quadratic, choosing the root consistent with positive drift.

1/32. Because the biased random walk moves one unit at a time, the hitting probability of -5 from 0 is h_1^5, where h_1 is the probability of ever going down one level. First-step analysis gives h_1 = 1/3 + (2/3) h_1^2, with roots 1 and 1/2, and the correct root for a walk with positive drift is 1/2. Hence the answer is (1/2)^5 = 1/32 = 0.03125. Equivalently (q/p)^{X_t} = (1/2)^{X_t} is a martingale, and optional stopping between -5 and an upper barrier pushed to infinity gives the same value. Drift makes deep drawdowns geometrically rare, so each extra level down costs a factor of 2.

Follow-up: What is the expected total number of times the walk visits 0?

Key concepts: biased random walk, first-step analysis, martingale, hitting probability.