Does the double-after-loss system beat a fair coin?

Question

You bet \1 on a fair coin. After a loss you double your next stake; after a win you pocket \1 of profit and reset to \1. Your bankroll is finite. Does this "martingale" betting system produce a positive expected profit? Explain.

Accepted Answer

Answer: No. Expected profit is exactly \0. Each individual bet is fair (expected value 0), so your bankroll is a martingale; by the optional stopping theorem, your expected wealth when you stop equals your starting wealth, whatever stopping rule you use. The system trades many small, likely wins for a rare, catastrophic loss, and the two cancel. Make it concrete. Suppose your bankroll allows n doublings before you are wiped out, that is, you can cover stakes 1, 2, 4, \dots, 2^{n-1}, a total of 2^n - 1. Two outcomes for one "attempt" (bet until a win or ruin): - With probability 1 - (	frac12)…

Does the double-after-loss system beat a fair coin?

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