The Gambler's Ruin formula
Given starting bankroll a, target b, and win probability p, the probability of ruin before reaching the target is P(ruin) = (1 โ (q/p)^a) / (1 โ (q/p)^b) where q = 1โp. For a fair game (p = 0.5), P(ruin) = 1 โ a/b.
- With 49% win probability, starting with $100 trying to reach $200: P(ruin) โ 67%
- The higher your win probability, the lower your ruin probability
- Larger bankroll relative to bet size dramatically reduces ruin probability
Practical implications
The formula shows that even slight edges against you (house edge) lead to near-certain ruin given enough time. Doubling your bankroll with a 49% game requires surviving a โ67% ruin probability. With a 45% game (like American roulette red/black), ruin probability is โ94%.
- Doubling strategy does not change long-run expectations โ only variance
- Having a larger bet-to-bankroll ratio increases ruin probability
- The infinite-play version: with any negative edge, P(eventual ruin) = 1