So I recently thought about the following game. You put down $x$, and you win $2x$ with probability $p < 0.5$. Clearly the expected value of this game is negative. Consider the following strategy. Let $b_i = 2^i$. Let $s$ be the first bet that you win. Then $b_s = 1 + \sum_{i < s} b_{i}$. Therefore you win one dollar. Its guaranteed that there exists $s$ where you win. I think it should not be possible intuitively. What is incorrect in this statement?
2026-03-28 11:29:08.1774697348
Guaranteed money with negative expected value bet
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Nothing is incorrect in the statement.
That is a well-known strategy for winning a small amount when the payout is twice the bet, no matter how big the bet is. And the problem lies with that clause, nobody offering such a game is going to allow the size of bets you quickly get to (they grow exponentially, if you lose the first 20 games, you'll have to bet more than 1.000.000 times the original bet, it might also be a problem to bring that much money). And the strategy only works if you're allowed to play until you win (and most places that offer this kind of game eventually closes for the night), else you end up losing a lot of money on the final series of games.
All in all: A well-known strategy, that's also very well analysed, and very impractical.