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TL;DR

A gambler with initial wealth a plays fair (or biased) coin flips until reaching 0 (ruin) or target N. In the fair game, the probability of ruin is \( (N-a)/N \). The expected duration is \( a(N-a) \).

By Valenke Exam Prep Team·Last updated 2026-06-03

Gambler's Ruin

A gambler with initial wealth a plays fair (or biased) coin flips until reaching 0 (ruin) or target N. In the fair game, the probability of ruin is \( (N-a)/N \). The expected duration is \( a(N-a) \).

Why it matters for interviews

One of the most classic probability problems, solvable by martingale methods or recurrence relations. It models drawdown risk in trading, the probability of a fund going bankrupt, and barrier problems in random walks.

Definition and Mathematical Foundation

A gambler with initial wealth a plays fair (or biased) coin flips until reaching 0 (ruin) or target N. In the fair game, the probability of ruin is \( (N-a)/N \). The expected duration is \( a(N-a) \).

Application in Quantitative Finance

One of the most classic probability problems, solvable by martingale methods or recurrence relations. It models drawdown risk in trading, the probability of a fund going bankrupt, and barrier problems in random walks.

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Frequently Asked Questions

How do you solve gambler's ruin with a biased coin?▼
For probability p of winning each bet, the ruin probability from wealth a is \( \frac{(q/p)^a - (q/p)^N}{1 - (q/p)^N} \) where \( q = 1-p \). For \( p < 0.5 \), ruin is almost certain for large N.
How does this relate to trading drawdowns?▼
A trading strategy with a slight edge (p > 0.5) can still experience long drawdowns. The gambler's ruin framework quantifies the probability of hitting a maximum drawdown before reaching a profit target.
What is the connection to martingales?▼
In the fair game, wealth is a martingale. By the optional stopping theorem, \( E[W_{\tau}] = a \). Since \( W_{\tau} \in \{0, N\} \), we get \( P(\text{ruin}) \cdot 0 + P(\text{win}) \cdot N = a \), yielding the ruin probability.