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

$$P= \begin{bmatrix} 0.3 & 0.2 & 0.5 \\ 0.5 & 0.1 & 0.4 \\ 0.5 & 0.2 & 0.3 \end{bmatrix} $$

and the initial distribution $p_0=0.5$ and $p_1=0.5$. what is the probability $\Pr(X_1=1,X_2=1,X_3=0)$ ?

I solved the problem as

$$\Pr(X_1=1,X_2=1,X_3=0)=\Pr(X_1=1)Pr(X_2=1\mid X_1=1)\Pr(X_3=0\mid X_2=1)$$ $$=p_1P_{1,1}P_{1,0}=(0.5)(0.1)(0.5)=0.025$$

But the answer is $0.0075$. I am not understanding where am I doing mistake?

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Recompute $Pr(X_1=1)$. It's not the same as $p_1$, which is $Pr(X_0=1)$.

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