13. Suppose that you are gambling against an infinitely rich adversary and at each stage you either win or lose 1 unit with respective probabilities p and 1 – p. Show that the probability that you eventually go broke is

$$

\begin{cases}

1 &\text{if } p \le \frac{1}{2}\\

\left(\frac{q}{p}\right)^i &\text{if } p > \frac{1}{2}

\end{cases}

$$

where q = 1 – p and where i is your initial fortune.