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In a betting game, you can win or lose a quantity $x$. The probability of winning a single bet is constant, $p$. You start with a wealth of $x$, which you bet in the first bet. What is probability of losing all the money, i.e. of ruin, in an infinite number of bets, as a function of $p$? I guess that the wording mentions "an infinite number of bets" in order to apply the Central Limit Theorem as the results from the single bets ($+x$ or $-x$) are iid random variables with finite mean and variance.

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What you have stated is a version of the classical gamblers ruin problem. There is a good article on that at wikipedia: https://en.wikipedia.org/wiki/Gambler's_ruin

Since you put all your wealth at stake in the first bet, it is clear that probability of ruin is larger than $1-p$. (That is the probability of loosing everything on round one, and you can loose it later also).

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