The doubling betting system

Question

A gambler bets \1 on a fair coin. Whenever he loses, he doubles his stake on the next toss (\2, then \4, then \8, and so on) and keeps doubling until he finally wins a toss, at which point he stops. When he wins, he recovers all his past losses plus a \1 profit. Since a fair coin lands his way eventually with probability 1, he claims he is guaranteed to end \1 ahead every time. This looks like a money machine. Explain why it is not.

Accepted Answer

It is true that he wins eventually with probability 1, and that the win nets exactly \1. The catch is in the resources the scheme silently demands. Consider the stake sizes. To survive a run of n losses in a row, the next bet must be 2^{n} dollars, and to place it he must be holding on the order of 2^{n} dollars in reserve. Losing streaks of any length occur with positive probability, and the expected size of the largest bet he must be prepared to make is infinite. In other words, the "guarantee" quietly assumes an unlimited bankroll and no table limit. Now put in a real, finite bankroll. The…

The doubling betting system

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