[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]

(The next three problems come from Feller.)

As in the text, assume that [math]M[/math] is a fixed positive integer.

• Show that if a gambler starts with an stake of 0 (and is allowed to have a negative amount of money), then the probability that her stake reaches the value of [math]M[/math] before it returns to 0 equals [math]p(1 - q_1)[/math].
• Show that if the gambler starts with a stake of [math]M[/math] then the probability that her stake reaches 0 before it returns to [math]M[/math] equals [math]qq_{M-1}[/math].

Notes