If $(X_n)_{n≥0}$ is a Markov chain on $S = {1, 2, 3}$ with initial distribution $α = (1/2, 1/2, 0)$ and transition matrix
$ \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix}, $
then $P(X_2 = 2) = ?$.
My solution:
$X_1 = \begin{bmatrix} 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix}$
$X_2 = \begin{bmatrix} 1/4&1/4&1/2 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} = \begin{bmatrix} 3/8&3/8&1/4 \end{bmatrix}$
So, $P(X_2=2) = 3/8$
Is this solution correct?
Why or why not?