<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

From where he stands, one step toward the cliff would send the
drunken man over the edge.  He takes random steps either towards
the cliff or away from the cliff.  At any step his probability of
taking a step away is $\frac{2}{3}$, or a step toward the cliff is
$\frac{1}{3}$.  What  is his chance of escaping the cliff?


\vspace{.25in}

{\bf Answer}

This problem is a simple random walk. with one absorbing barrier.
It is equivalent to a gambler's ruin problem, played against an
infinitely rich opponent with $$ z = 1 \hspace{.2in} p =
\frac{2}{3} \hspace{.2in} {\rm and} \hspace{.2in} q =
\frac{1}{3}$$ The probability of ruin is $\ds \left( \frac{q}{p}
\right) ^z = \frac{1}{2}$ in this case.





\end{document}
</pre></body></html>