The transition probability matrix of a markov chain with $S = \{0, 1, 2\}$ is
\begin{bmatrix} .2 & .8 & 0 \\ 0 & .3 & .7 \\ .9 & 0 & .1 \\ \end{bmatrix}
How do I compute $E(X_1X_2\;|\;X_0=0)$?
The answer is $1.36$, but I have no idea how to get to that point. I tried computing $E(X_1|X_0=0)$ and $E(X_2|X_0=0)$ separately and finding the product, but I got a different answer.
You are at state $0$ at time $0$, hence you are either at state $0$ or $1$ at time $1$.
If $X_1=0$, then $X_1X_2=0$.
If you are at state $1$ at time $1$, at time $2$, you are either at state $1$ or at state $2$.
So it suffices to compute the probability of taking path $(0,1,1)$ [and then $X_1X_2=1$] and $(0,1,2)$ [and then $X_1X_2=2$].
Can you complete the rest?
Remark: By attempting the question, you learned that $X_1$ and $X_2$ are not independent given $X_0$.