For a finite state Markov process and a finite collection $\{ Γ_k, k \in K \}$ of subsets of its state space, let $τ_k$ be the first time the process visits the set $Γ_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times $\{ τ_k, k \in K\}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.