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A Markov chain $X_0$, $X_1$, $X_2$, ... has the transition probability matrix

$$ P = \left[ \matrix { 0.3&0.2&0.5 \\ 0.5&0.1&0.4 \\ 0&0&1 } \right] $$

and is known to start in state $X_0$ = 0. Eventually, the process will end up in state 2. What is the probability that when the process moves into state 2, it does so from state 1?

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Let $\tau = \min\{n: X_{n+1}= 2\}$ and $p_{i,j} = P(X_{\tau}=i|X_{0}=j)$. Observe that:

$$p_{0,0} = P(X_{\tau}=0|X_{0}=0) = \sum_{k=0}^{2}{P(X_{\tau}=0,X_{1}=k|X_{0}=0)} =$$

$$=\sum_{k=0}^{2}{P(X_{1}=k|X_{0}=0)P(X_{\tau}=0|X_{1}=k,X_{0}=0)}$$

Do you know $P(X_{1}=k|X_{0}=0)$? Can you simplify $P(X_{\tau}=0|X_{1}=k,X_{0}=0)$ in terms of $p_{0,0}$ and $p_{0,1}$? Do the same steps for $p_{0,1}$ and solve the system of equations.

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