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How do you prove that a "system" where you vary your bet size from one trial to the next cannot help you beat an unfavorable game? For concreteness, just assume the game is roulette and you are betting on "red" each time. Also assume that the bet sizes are in discrete units like dollars, and that there's a minimum and a maximum bet limit. You are free to vary your bet size between those limits however you like based on past outcomes.

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    $\begingroup$ math.utah.edu/~levin/M5040/mg.pdf $\endgroup$ – Lorenzo Jun 18 at 4:10
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    $\begingroup$ In roulette, every bet has a negative expectation. $\endgroup$ – Lord Shark the Unknown Jun 18 at 6:08
  • $\begingroup$ I believe that is correct $\endgroup$ – Willie Betmore Jun 18 at 23:51
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    $\begingroup$ @Lorenzo thanks for those notes. I assume that to make that analysis apply to an unfavorable game, you just replace some equal signs with less than signs. Is there a way to go from the conclusion that, on average, things go poorly, to the more interesting conclusion that, almost surely, things will go poorly if you keep playing? $\endgroup$ – Willie Betmore Jun 19 at 3:12
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    $\begingroup$ If you can formulate a precise question capturing your intuition, try opening a new question here. There are a lot of theorems about the convergence of martingales and stopping times (when you run out of money), and so on... $\endgroup$ – Lorenzo Jun 19 at 4:09

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