Consider the Markov chain $(X_n)$ consisting of the three states $\{1,2,3\}$ and having transition probability matrix $$\left (\begin{matrix} 1/3 & 2/3 & 0\\ 1/2 & 0 & 1/2 \\0 & 0 & 1 \end{matrix} \right ).$$ The distribution of $X_0$ is $(0.1;0.3;0.6)$. Find $$Pr(X_1 \in \{1,2\}|X_0 \in \{1,3\}).$$
2026-04-24 03:48:32.1777002512
Markov chains and conditional probability on subset of state space
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$$P(X_1\in\{1,2\} |X_0\in \{1,3\}) = \frac{P(X_1\in\{1,2\}, X_0\in \{1,3\})}{P(X_0 \in \{1,3\})} = \frac{P(X_1\in\{1,2\}|X_0 = 1)P(X_0=1) + P(X_1\in\{1,2\}| X_0=3)P(X_0=3)}{P(X_0 = 1) + P(X_0=3)} = \dotsi $$ Does this help?