[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]

Toss a fair die repeatedly. Let [math]S_n[/math] denote the total

of the outcomes through the [math]n[/math]th toss. Show that there is a limiting value for the proportion of the first [math]n[/math] values of [math]S_n[/math] that are divisible by 7, and compute the value for this limit. Hint: The desired limit is an equilibrium probability vector for an appropriate seven state Markov chain.