[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that a gambler starts with a stake of 0 dollars.

• Show that the probability that her stake never reaches [math]M[/math] before returning to 0 equals [math]1 - p(1 - q_1)[/math].
• Show that the probability that her stake reaches the value [math]M[/math] exactly [math]k[/math] times before returning to 0 equals [math]p(1-q_1)(1 - qq_{M-1})^{k-1}(qq_{M-1})[/math]. Hint: Use Exercise.