[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]

In the text, it was shown that if [math]q \lt p[/math], there is a positive probability that a gambler, starting with a stake of 0 dollars, will never return to the origin. Thus, we will now assume that [math]q \ge p[/math]. Using Exercise, show that if a gambler starts with a stake of 0 dollars, then the expected number of times her stake equals [math]M[/math] before returning to 0 equals [math](p/q)^M[/math], if [math]q \gt p[/math] and 1, if [math]q = p[/math]. (We quote from Feller: “The truly amazing implications of this result appear best in the language of fair games. A perfect coin is tossed until the first equalization of the accumulated numbers of heads and tails. The gambler receives one penny for every time that the accumulated number of heads exceeds the accumulated number of tails by [math]m[/math]. The ‘fair entrance fee’ equals 1 independent of [math]m[/math].”)