# 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 == $\infty$?

• Going left or right has same probability? – drhab Jan 13 '15 at 10:06
• @drhab - yes, thanks. – R B Jan 13 '15 at 10:10
• @drhab - although if we can get the distribution when going lower is w.p. $p\geq \frac{1}{2}$ it's also interesting. – R B Jan 13 '15 at 10:12

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.