# Why does this probability equivalence of events hold?

$P(X_0 = j, X_m \ne j, 1 \le m \le n-1) = P(X_m \ne j, 1 \le m \le n-1) - P(X_m \ne j, 0 \le m \le n-1)$

Where

$\{X_n\}$ is an irreducible Markov Chain with a finite state space.

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This is basic probability; it has nothing to do with Markov chains. Define $A=(X_0=j)$ and $B=\cap_{m=1}^{n-1}(X_m\neq j)$. Then your equation simply says $$\mathbb{P}(A\cap B)=\mathbb{P}(B)-\mathbb{P}(\bar A\cap B).$$ This is true because $B$ is the disjoint union of $A\cap B$ and $\bar A\cap B$, where I'm using $\bar A$ to denote the complement of $A$.