Conditional independence of events

Question

Let $A$, $B$ and $C$ be arbitrary events defined in the same space.

If $A$ conditionally in $B$ is independent of $C$; i.e., $\mathbb{P}(A | B \cap C) = \mathbb{P}(A | B)$, is it true that $C$ conditionally in $B$ is independent of $A$; i.e., $\mathbb{P}(C | B \cap A) = \mathbb{P}(C | B)$?

Answer 1

The answer to your question is NO. Let me give you a counterexample:

If $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space and we set $A=\varnothing$ and $B=C=\Omega$, then we have

$$\mathbb{P}(A\,\vert B\cap C)=\mathbb{P}(A\,\vert B)=0.$$

On the other hand, we have $\mathbb{P}(C\,\vert B)=1$ but $\mathbb{P}(C\,\vert B\cap A)$ is not defined.

Answer 2

Hint: is $$\frac{P(ABC)}{P(BC)}=\frac{P(AB)}{P(B)}$$ same as $$\frac{P(CBA)}{P(BA)} = \frac{P(BC)}{P(B)}$$ ?