Is this Markov Chain calculation correct?

Question

$S=\{1,2\}$
$\alpha = (1/2, 1/2)$
$P= \begin{bmatrix} 1/2&1/2\\ 0&1\end{bmatrix} $

Find

• $P(X_1=1 | X_0=1)$

Given solution:

$P(X_1=1 | X_0=1) = \frac{P(X_1=1 , X_0=1)}{P(X_0=1)} = \frac{P(X_1=1) \cdot P( X_0=1)}{P(X_0=1)} = 0.5 * 0.5 / 0.5 = 0.5$

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To me, this is incorrect.

Because, $P(X_1=1) = 0.25$ .... as $\alpha \cdot P = (0.25, 0.75)$

$P(X_0=1) = 0.5$ is correct as the 1st element of $\alpha = (1/2, 1/2)$ is $0.5$.

So, the answer should be $0.25$.

Answer 1

I can’t quite agree with the provided solution but the question can directly be answered using your probability matrix, since $P_{ij}=P(X_{k+1}=i\vert X_k=j)$ for any non-negative $k$.