Suppose we are flipping a coin with probability $p$ of coming up heads and $q$ of coming up tails. We start with $n$ dollars, and the game is over when we either lose all our money or win $m$ dollars. Each toss wins (if heads) or loses (if tails) $\\\$1.$ We play the game for $t$ turns (or until we hit one of the barriers). Question: what is our expected capital at the end of the game? Obviously, if $p>q,$ and $t\gg 1,$ we asymptote to $m+n,$ and if $t < \min(m, n),$ the expectation is simply $t(p-q).$ But in between it seems a bit less obvious.

Suppose we are flipping a coin with probability $p$ of coming up heads and $q$ of coming up tails. We start with $n$ dollars, and the game is over when we either lose all our money or win $m$ dollars. Each toss wins (if heads) or loses (if tails) $\\\$1.$ We play the game for $t$ turns (or until we hit one of the barriers). Question: what is our expected capital at the end of the game? Obviously, if $p>q,$ and $t\gg 1,$ we asymptote to $m+n,$ and if $t < \min(m, n),$ the expectation is simply $n+t(p-q).$ But in between it seems a bit less obvious.