How to prove that conditional independence does not imply independence? I am trying to prove that conditional independence does not imply independence, ie that $P(A|C)P(B|C)=P(A \cap B|C) \nRightarrow P(A \cap B)=P(A)P(B)$
I guess I need a counter-example but I am struggling to find a way  of homing in onto one. 
So far I have tried drawing Venn diagrams, and I can see that there is no reason why the sizes of the relevant intersections should multiply as implied by the above, but I am not sure how to proceed from there.

Let $A,B$ be events that are not independent.
Observe that $P(A\mid A)P(B\mid A)=P(B\mid A)=P(A\cap B\mid A)$ showing that there is conditional independence wrt $A$.
