Suppose states $i, j$ are symmetric in a Markov Chain, i.e.:
$P(T_j<T_i\mid X_0=i)=P(T_i<T_j\mid X_0=j):=\theta$,$\quad$ where $T_i=\min\{n\geq1:X_n=i\}$.
Denote $N$ as the number of visits to state $j$ before visiting state $i$. I cannot see why $P(N\geq k\mid X_0=i)=\theta(1-\theta)^{k-1}$ holds.
I think it should be $P(N=k\mid X_0=i)=\theta(1-\theta)^{k-1}$, because for $k-1$ times, the chain doesn't arrive state $j$, and at the $k$th time, the chain arrives at $j$.