Risk of ruin with a small edge

Question

You make even-money bets that you win with probability $p = 0.53$ (edge in your favor). You stake $1$ unit per bet and start with a bankroll of $n = 20$ units, playing against a market with effectively unlimited capital.

What is your probability of eventual ruin? How does it change if you bet smaller (more units of bankroll)?

Accepted Answer

This is the gambler's-ruin problem against an infinitely deep opponent. For an even-money bet with win probability p q = 1-p, the probability of ever being ruined starting from n units is P(	ext{ruin}) = \left(\frac{q}{p}ight)^{n}. With p = 0.53, q = 0.47: \frac{q}{p} = \frac{0.47}{0.53} \approx 0.887, \qquad P(	ext{ruin}) = 0.887^{20} \approx e^{20\ln 0.887} = e^{-2.40} \approx 0.091. So about a 9\% chance of going broke despite a real edge, and a 91\% chance of running your bankroll to infinity. A positive edge does not guarantee survival; variance can still ruin you before the edge pla…

Risk of ruin with a small edge

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