A Note on Exact State Visit Probabilities in Two-State Markov Chains

In this note we derive the exact probability that a specific state in a two-state Markov chain is visited exactly $k$ times after $N$ transitions. We provide a closed-form solution for $\mathbb{P}(N_l = k \mid N)$, considering initial state probabilities and transition dynamics. The solution corrects and extends prior incomplete results, offering a rigorous framework for enumerating state transitions. Numerical simulations validate the derived expressions, demonstrating their applicability in stochastic modeling.