Markov chain probability that a state changes

Question

For the Markov chain given below what is the best way to find the probability P{$x_n = x_{n-1}$} and P{$x_n \neq x_{n-1}$}

The transition matrix of the chain is \begin{array}{} 1/3 & 2/3 & 0 \\ 0 & 2/3 & 1/3 \\ 0 & 2/3 & 1/3 \end{array}

Answer 1

For every state $x$, $P(x_1=x\mid x_0=x)=\frac13$, hence $P(x_n=x_{n-1})=\frac13$ and $P(x_n\ne x_{n-1})=\frac23$, for every initial distribution and every $n$.