Gambler's Ruin

Problem Statement#

The Question#

Interview question: Player A starts with $k and Player B starts with $\$(n-k)$, for a combined total of $n$ dollars. Each round, A wins $1 from B with probability $p$, and loses $1 to B with probability $q = 1-p$. They keep playing until one player is broke. What is the probability that Player A is ruined?

This problem shows up at virtually every top quant firm, Jan...