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## Main Question or Discussion Point

Does the law of large numbers really apply directly to betting systems? For example, in American roulette the house advantage or "edge" is 5.26%, and smart players know that, as a consequence of the law of large numbers, you will lose 5.26 cents of every dollar bet in the long run. This is supposed to apply no matter what ingenious betting system you use, as long as there is a maximum bet and you keep playing indefinitely.

The statement of the strong law of large numbers usually applies to a sum of independent random variables. But the winnings from each trial are not independent, even though the trial outcomes themselves are. This is because the bet sizes on different trials are not independent for most any "system". So how do you know that the law of large numbers applies to the total winnings after n trials?

The statement of the strong law of large numbers usually applies to a sum of independent random variables. But the winnings from each trial are not independent, even though the trial outcomes themselves are. This is because the bet sizes on different trials are not independent for most any "system". So how do you know that the law of large numbers applies to the total winnings after n trials?