Chapter 9.1, Problem 51E

Problems 61-70 refer to the following transition matrix $P$ and its powers

$\begin{array}{l}\begin{array}{ccc}A& B& C\end{array}\\ P=\begin{array}{c}A\\ B\\ C\end{array}\left[\begin{array}{ccc}.6& .3& .1\\ .2& .5& .3\\ .1& .2& .7\end{array}\right]\end{array}$ $\begin{array}{l}\begin{array}{ccc}A& B& C\end{array}\\ P^2=\begin{array}{c}A\\ B\\ C\end{array}\left[\begin{array}{ccc}.43& .35& .22\\ .25& .37& .38\\ .17& .27& .56\end{array}\right]\end{array}$ $\begin{array}{l}\begin{array}{ccc}A& B& C\end{array}\\ P^3=\begin{array}{c}A\\ B\\ C\end{array}\left[\begin{array}{ccc}.35& .348& .302\\ .262& .336& .402\\ .212& .298& .49\end{array}\right]\end{array}$

Find the probability of going from state $C$ to state $A$ in three trials.