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gambler's ruin is the principal of probability that states that if a gambler were to start out with any finite sum of money and gambled for and infinite ammount of time, no matter how good the odds were, eventually, he or she would run out of money.

In a normal casino, where the payout multipliers are less than the odds of winning (due to house percentage), the Gambler's Ruin is close enough to fact to be treated as such. But in a fair game, the odds are even between you getting ruined before and playing successfully up to N plays (because at any point, you're just as likely to have a winning as a loss).

In the more general case, if each play is in your favor, there is a nonzero positive probability of playing infinitely without being ruined. If each play is not in your favor (the typical casino case), then with probability approaching 1 you get ruined as you play to infinity.

It's possible, of course, to play to an arbitrarily high number of plays without getting ruined even when the odds are against you, which is where the "Law" of Averages falls down (the LoA treats independent events as dependent ones, which is what people do when they say they're "due" to win after a long stretch of poor luck).

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