# Gambler's Ruin with no set target for win

I have been presented with the following probability question: A compulsive gambler is never satisﬁed. At each stage he wins $€1$ with probability $p$ and loses $€1$ otherwise. Find the probability that he is ultimately bankrupted, having started with an initial fortune of $€k$.

I am familiar with the gambler's ruin problem where a gambler continues to play until he wins or is bankrupted, and also the problem of the drunk man atop the cliff who steps left or right until he makes it home or falls from the cliff.

My question is this: in the problem I have been presented with the gambler does not have a target that he must reach to "win" but plays compulsively until he loses. Does this mean that in this version of the problem the probability of bankruptcy is one? This seems correct intuitively but I cannot seem to prove it. Any help would be appreciated.

## 1 Answer

You can think about this as starting with the original Gambler's Ruin problem, but pushing the target N out to infinity. For $p \leq 0.5$ the probability of eventually going bankrupt is indeed 1, but $p > 0.5$ gives a probability of $1 - \big(\frac{1-p}{p}\big) ^ k$ of never going bankrupt.