Duration of a Gambler's Ruin game against an opponent with infinite credit A gambler enters the casino with $x\$$ in his pocket and sits on some table. At each iteration he bets $1\$$ and either wins $1\$$ with probability $p\leq\frac{1}{2}$ or loses $1\$$. Assuming that the casino has unlimited credit, it's simple to see that the gambler will eventually get bankrupt. > **How is the time till bankruptcy distributed?** > > **Is the expected time till bankruptcy == $\infty$?**

> How is the time till bankruptcy distributed?

This is an application of the Hitting Time Theorem (see, e.g. here (Theorem 1) or pg. 79 of Grimmett and Stirzaker).

$$P(\text{Ruined at game $n$ starting with $\$x$}) = \dfrac{x}{n}\binom{n}{(n-x)/2}p^{(n-x)/2}q^{(n+x)/2}.$$

> Is the expected time till bankruptcy $= \infty$?

Yes, if $p\geq q$. Otherwise, it is

$$\dfrac{x}{q-p}.$$

Ref: e.g. Section 2.1.2 of here or G&S pg. 74. In both references take the limit as casino's fortune approaches $\infty$ because they assume a finite casino amount.