[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]

A certain experiment is believed to be described by a two-state Markov chain with the transition matrix [math]\mat{P}[/math], where

[[math]] \mat {P} = \pmatrix{ .5 & .5 \cr p & 1 - p} [[/math]]

and the parameter [math]p[/math] is not known. When the experiment is performed many times, the chain ends in state one approximately 20 percent of the time and in state two approximately 80 percent of the time. Compute a sensible estimate for the unknown parameter [math]p[/math] and explain how you found it.