# Probability - Biased coin — Betting game

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.

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).