Having a two-state Markov chain with the symmetric transition matrix: \begin{pmatrix} 1-p&p\\ p&1-p \end{pmatrix} The states are 1 and 2, let $n_{2\rightarrow1}(t)$ be the number of transitions from state 2 to state 1 up to time t (after t total transitions).
I was wondering if the ratio $\frac{n_{2\rightarrow1}(t)}{t}$ converges to a specific value or no.
I tried to calculate the expected value of $n_{2\rightarrow1}(t)$: $\sum_{n=1}^{t}np_{n}$, where $p_n$ is the probability of finding n $2\rightarrow1$ transitions up to t total transitions, but i'm not sure that this is the correct form to find the expectancy, any help would be really appreciated.