Chapter 9, Problem 12RE

Given the transition matrix

$\begin{array}{l}\begin{array}{cc}A& B\end{array}\\ P=\begin{array}{c}A\\ B\end{array}\left[\begin{array}{cc}.4& .6\\ .9& .1\end{array}\right]\end{array}$

find the probability of

(A) Going from state $A$ to state $B$ in two trials

(B) Going from state $B$ to state $A$ in three trials