[Pokey]

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If you just double your bet till you win and then go back to one each time tell me why that does not make you money. If there is no max bet and you have all the money in the world.

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You're right, so long as you make those two assumptions. If either of those assumptions does NOT hold true, however, then you can lose your entire bankroll, and you'll do so more often than you'll double your money. In the long run, your expectation is negative because your gambles are all negative. It's only in the infinite situations that you can make money with this strategy.

Quick example: say you start with $255 and you use a martingale betting strategy where you're going to win 49% of your bets and lose 51% of your bets. You play until you either double your money or wipe yourself out. The $255 lets you bet $1, then $2, then $4, then $8, $16, $32, $64, and $128; at that point, you're out of cash. The odds of hitting a losing streak of 8 in a row (bankruptcy) is going to be (0.51)^8 = 0.46%. In order to double your money, you'd have to avoid that bad situation 255 times in a row. The odds of dodging a 0.46% event 255 times in a row is (1-0.0046)^255 = 31%. In other words, you'd lose $255 69% of the time and win $255 31% of the time. (The math isn't *quite* perfect because you'd still have a non-zero bankroll if your catastrophic loss came after the first time, but your bankroll would be smaller, giving you an even lower chance of rebuilding; suffice it to say it's in the ballpark.) Even with a very minor -EV betting situation, martingale strategies are losers in the long run. Of course, we should have already known that, since ALL -EV bets are losers in the long run.