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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.
Formula
- P
- 0 — 0
- N
- N value — Variable used in the calculation
- p
- 0 — 0
Guida passo passo
- 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
Esempi risolti
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What is Gambler Ruin?
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. Use this calculator for accurate, instant results.
How accurate is the Gambler Ruin calculator?
The calculator uses the standard published formula for gambler ruin. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.
What units does the Gambler Ruin calculator use?
This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.
What formula does the Gambler Ruin calculator use?
The core formula is: P(ruin | start at k) = (rᵏ − rᴺ) / (1 − rᴺ) where r = (1−p)/p. Each step in the calculation is shown so you can verify the result manually.
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