Chapter 10.2, Problem 37E

To determine

To show: $\left[\begin{array}{l}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$11$}\right.\\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$11$}\right.\\ \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$11$}\right.\\ 0\\ 0\end{array}\right]$and $\left[\begin{array}{l}0\\ 0\\ 0\\ \raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$17$}\right.\\ \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$17$}\right.\end{array}\right]$ are steady state vectors of the Markov chain $\left\{1,2,3,4,5\right\}$ with transition matrix $\left[\begin{array}{ccccc}1/4& 1/3& 1/2& 0& 0\\ 1/4& 1/3& 1/4& 0& 0\\ 1/2& 1/3& 1/4& 0& 0\\ 0& 0& 0& 1/3& 3/4\\ 0& 0& 0& 2/3& 1/4\end{array}\right]$ .

Is the chain is equally like to begin in each of states, what is the probability of being in state $1$ after many steps.