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Finding the value of ##P(X_3 = 1|X_1 = 2) = ?## in a Markov Chain

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user366312

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Problem Statement
If ##(X_n)_{n≥0}## is a Markov chain on ##S = \{1, 2, 3\}## with initial distribution ##α = (1/2, 1/2, 0)## and transition matrix

## \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix},##

then ##P(X_3 = 1|X_1 = 2) = ?##.
Relevant Equations
Markov Chain
##P^2=\begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix}=
\begin{bmatrix}
1/2 & 1/4 & 1/4\\
1/4 & 1/2 & 1/4\\
1/4 & 1/4 & 1/2
\end{bmatrix}##

So, ##P(X_3 = 1|X_1 = 2) = 1/4##.

Is this solution correct?
 

Orodruin

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Yes.
 

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