Let $(X_n)_{n\ge0}$ be a Markov chain with the state space $S$ and Transition Kernel $P$. Let $A\subset S$ be a subset of interest, and $T^A$ the first time that the MC hits the set A: $$T^A(\omega)=\inf\{n\ge0:X_n(\omega)\in A\}$$

For each $i\in S$, denote the probability of hitting $A$ starting from position $i$ by $$h^A_{i}=P(T^A<\infty|X_0=i)$$where $T^A$ is the hitting time for event $A$. For simplicity, we will write $P_i$ for $P(\cdot|X_0=i)$.

For $i\in S$, the expected time of hitting $A$ is $$\begin{align*} \kappa_{i}^{A}&=E_{i}[T^{A}]\\ &=E[T^{A}|X_{0}=i] \end{align*}$$where $T^{A}$ is the hitting time of event $A$. By definition we have $$\kappa_{i}=\sum\limits_{n<\infty}nP_{i}(T^{A}=n)+\infty*P_{i}(T^{A}=\infty)$$ If the probability of not hitting the set $A$ from state $i$ is positive, then $\kappa_{i}=\infty$. - i.e. if hitting probability $h_{i}=1$ then $\kappa_{i}<\infty$ or “if we’re for sure hitting A then we have a expected hitting time” - i.e. if hitting probability $h_i<1$ then $\kappa_{i}=\infty$ or “if we’re not hitting A for sure then expected hitting time is infinite”