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

#### user366312

Gold Member
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?

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Yes.

#### user366312

Gold Member
i have another thread. kindly see that also.

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

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