I have to show that the equation
$P(X_u=b|X_t \in A, X_s=a)= P(X_u=b|X_t\in A)$
is wrong for a time-homogeneous Markov chain $(X_k)_{k\in \mathbb N_0}$ on a finit space $E$ with $s<t<u \in \mathbb N_0$, $a,b \in E$ and for all $A \subset E$. Furthermore there is a hint with the following graph:
which shows a markov chain with its stochastic matrix
$ \begin{bmatrix} 0 & 1/2 & 1/2 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 1& 0 & 0 & 0\\ 0&0&0&0&1 \end{bmatrix} $ .
I somehow can't find a counterexample and don't really know how i can calculate $P(X_u=b|X_t \in A, X_s=a)$. I hope someone can help me.