🎰Gambler's Ruin Calculator
e.g. 0.49
The Gambler's Ruin problem asks: starting with k units, betting 1 unit per round with win probability p, what is the probability of reaching target N before going bankrupt? The answer reveals that even with a small house edge, the gambler is almost certain to be ruined eventually — a mathematical proof of why gambling systems cannot overcome negative expected value.
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Tip: The lesson of Gambler's Ruin: the larger your target relative to your bankroll, and the worse your per-bet edge, the more certain your ruin. No betting system (Martingale, Fibonacci, etc.) can change the underlying mathematics.
- 1P(ruin | start at k) = (rᵏ − rᴺ) / (1 − rᴺ) where r = (1−p)/p
- 2Fair game (p = 0.5): P(ruin) = 1 − k/N
- 3Unfair game (p < 0.5): ruin probability rapidly approaches 1 as N → ∞
- 4Expected duration: k(N−k)/(1−2p)² for p ≠ 0.5
Start £100, target £200, win prob 49% (slight house edge)=P(ruin) ≈ 55%, P(win) ≈ 45%
Start £100, target £10,000, win prob 49%=P(ruin) > 99.99%Huge targets are nearly impossible
Fair game (50/50), start £100, target £200=P(ruin) = 50%
| Win Probability | Start £100 Target £200 | Start £100 Target £1000 |
|---|---|---|
| 50% (fair) | 50.0% | 90.0% |
| 49% (roulette-like) | 55.0% | 99.3% |
| 47.4% (roulette, US) | 61.2% | >99.9% |
| 45% (heavy edge) | 71.4% | >99.9% |
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