Fred is playing a game that costs $\$1$ per round. Each round, he wins with probability $p$ (where $0 1$), but he does not get his $\$1$ entry fee back. If he loses, he simply loses his $\$1$. Fred starts with $\$s$ in his bank (an integer with $s \geq 1$). He keeps playing until he goes bankrupt (…

Probability of Exactly One Win Before Bankruptcy

Probability · Medium · Free problem

Fred is playing a game that costs $\

$ per round. Each round, he wins with probability $p$ (where $0 < p < 1$) and loses with probability

- p$, independently. If he wins a round, he receives $\$k$ (an integer with $k > 1$), but he does not get his $\

$ entry fee back. If he loses, he simply loses his $\

$. Fred starts with $\$s$ in his bank (an integer with $s \geq 1$). He keeps playing until he goes bankrupt (hits $\$0$). Find the probability that Fred wins exactly once before going bankrupt. Evaluate your answer for $s = 3$, $k = 4$, and $p = 1/3$. Round to the nearest ten-thousandth.

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