Consider a two-state Markov chain with the corresponding transition matrix $$ P = \begin{bmatrix} \frac{5}{8}&\frac{3}{8} \\\frac{3}{8}&\frac{5}{8}\\ \end{bmatrix}. $$ Given an equal chance of starting at either state, how many stops will there be on average at the first state (corresponding to the first entry of the probability vector) after $n$ stages/steps?
The initial condition for this problem is given by $\mathbf{p}(0) = (1/2, 1/2)$ but I cannot figure out how to proceed from here. Any hints on where to start are appreciated.