The problem. A gambler starts with wealth between 0 and an intended target, and bets until either reaching the target or going broke. The gambler’s ruin problem asks for the probability of winning — the classical entry point into random walk theory, and the ancestor of ruin theory in actuarial mathematics.
What the report covers. A unified, self-contained treatment across three settings. For the simple symmetric walk, conditioning on the first step (a difference equation) and the optional sampling theorem both give the exact answer: the winning probability scales linearly with the starting position. For the spread-out model with bounded steps, the “overshoot” past the boundary breaks that exact identity, but the martingale property still yields tight bounds and the same linear scaling. For the finite-variance case, linear scaling survives even when the steps are unbounded. The report closes with three extensions: the link between the discrete Laplacian and potential theory, the Poisson equation for expected hitting time, and the Brownian-motion limit.
Context. An expository study report advised by Prof. Yuki Chino (Department of Applied Mathematics, NYCU); results follow Lawler & Limic (2010) and Koralov & Sinai (2007).