Gambler's Ruin

Tap a term to see what it means, or .

Calculate

Example: in European roulette, betting on red wins with chance 18/37 ≈ 48.65%. With $50, betting $1 at a time, what is the chance of doubling your money before losing it all? Change the values to work out your own case.

Starting capital i Target N Chance of winning a round p (%)

Step by step

What each term means

Chance of Reaching the Target (Pᵢ): Starting with i, the probability of reaching N before reaching zero, winning or losing 1 per round.
Numerator: q/p is the ratio of losing to winning a round. With q/p > 1 the game is unfavourable, and (q/p)ⁱ grows fast with the capital.
Denominator: The same calculation for the target N. Since N > i, in an unfavourable game the denominator is far larger than the numerator, and the chance shrinks. With p = q, the formula becomes i/N.