I want to find a counterexample to verify that the following is false $$ P(A \cap B \mid C) = P(A \mid C) \, P(A \cap C) $$ So I thought I would take $A=C$ and then $$ P(A \cap B \mid C) = \frac{P(C \cap B)}{P(C)}, $$ and if, for example, $C \cap B = \varnothing$ with $C \neq \varnothing$, then $$ P(A \cap B \mid C) = 0, $$ but $$ P(A \mid C) \, P(A \cap C) = P(A \cap C) \neq 0. $$
Can this counterexample be useful? or is there a better one? thanks!