[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]

Show that [math]w_x[/math] of Exercise satisfies the following conditions:

• [math]w_x = pw_{x + 1} + qw_{x - 1}[/math] for [math]x = 1[/math], 2, ..., [math]T - 1[/math].
• [math]w_0 = 0[/math].
• [math]w_T = 1[/math].

Show that these conditions determine [math]w_x[/math]. Show that, if [math]p = q =1/2[/math], then

[[math]] w_x = \frac xT [[/math]]

satisfies (a), (b), and (c) and hence is the solution. If [math]p \ne q[/math], show that

[[math]] w_x = \frac{(q/p)^x - 1}{(q/p)^T - 1} [[/math]]

satisfies these conditions and hence gives the probability of the gambler winning.