Let $(X_{t \in \mathbb{N_0}})$ be a Markov Chain with states $E$. Let $A,B \subseteq E$ with $x_0,x_1 \in E$.
Prove the following:
- $P (X_2 ∈ B|X_1 = x_1 , X_0 ∈ A) = P (X_2 ∈ B|X_1 = x_1 )$.
- $ P(X_2 ∈ B|X_1 ∈ A, X_0 = x_0 ) = P (X_2 ∈ B|X_1 ∈ A) $
I believe the $1^{st}$ is true but for the $2^{nd}$ I came with a counter example (hopefully right one).
$A=\{x_0,x_1\}, B=\{x_2\}$
$$P(X_2 \in \{x_2\} | X_1 \in \{x_0,x_1\}, X_0=x_0)=0$$ $$P(X_2 \in \{x_2\} | X_1 \in \{x_0,x_1\})=1/2$$