The Gambler's Ruin problem models a bettor's fortune as a one-dimensional random walk with absorbing boundaries at $0$ and $N+M$, where each step moves up by one with probability $p$ and down by one with probability $1-p$, independently of prior steps (a martingale). The probability of reaching the upper boundary before the lower one, and the expected number of steps until absorption, are derived by setting up linear recurrences in the starting capital and solving their characteristic equations — yielding a closed form via distinct roots when $p \neq 1/2$ (biased walk) and via a repeated root when $p = 1/2$ (unbiased/fair walk), with the qualitative behavior of the two cases diverging sharply because of this root multiplicity. The result belongs to probability theory's treatment of random walks and stopping times, illustrating how a deterministic "drift" term asymptotically dominates random fluctuations ("swings") that grow only as the square root of the number of steps.
The Gambler's Ruin problem models a bettor's fortune as a one-dimensional random walk with absorbing boundaries at $0$ and $N+M$, where each step moves up by one with probability $p$ and down by one …