[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]

Assume in the gambler's ruin problem that [math]p = q = 1/2[/math].

• Using Equation, together with the facts that [math]q_0 = 1[/math] and [math]q_M = 0[/math], show that for [math]0 \le z \le M[/math],
  [[math]] q_z = {{M - z}\over M}\ . [[/math]]
• In Equation, let [math]p \rightarrow 1/2[/math] (and since [math]q = 1 - p[/math], [math]q \rightarrow 1/2[/math] as well). Show that in the limit,
  [[math]] q_z = {{M - z}\over M}\ . [[/math]]
  Hint: Replace [math]q[/math] by [math]1-p[/math], and use L'Hopital's rule.