Let $\{X_n:n\geq 0\}$ be a Markov chain with three states $1,2,3$ and transition matrix $[p_{ij}]=\begin{bmatrix} 3/4 & 1/4 & 0 \\ 1/4 & 1/2 & 1/4 \\ 0 & 3/4 & 1/4 \end{bmatrix}$ with initial distribution $p_i=P(X_0=i)=\frac{1}{3},i=1,2,3$. Find $P(X_3=2,X_2=3,X_1=2,X_0=3)$.
My Attempt:
$P(X_3=2,X_2=3,X_1=2,X_0=3)$ $= P(X_0=3)P(X_1=2|X_0=3)P(X_2=3|X_1=2)P(X_3=2|X_2=3)$ $=\frac{1}{3}\frac{3}{4}\frac{1}{4}\frac{3}{4}$
Is my attempt correct? I was a bit confused because someone suggested me that I just neglect whatever happened before $X_2=3$ because it is a Markov chain. But I feel that would only be correct if they had asked for $P(X_3=2|(X_2=3,X_1=2,X_0=3))$. Here we don't know for certain whether the chain had actually passed through $X_0=3,X_1=2$ and $X_2=3$.
Good job. Your work is correct.
Your concern is right too. We cannot neglect wharver happens before $X_2=3$.
If we were neglect those terms, it would have been an overestimation \begin{align}&P(X_3=2, X_2=3, X_1=2, X_0 = 3) \\&= P(X_3=2|X_2=3, X_1=2, X_0 = 3)P(X_2=3, X_1=2, X_0 = 3)\\ &=P(X_3=2|X_2=3)P(X_2=3, X_1=2, X_0=3) \\&< P(X_3=2|X_2=3)\end{align}