Is Independence Stable under Intersections?

Question

Let $A,B,C$ be events. If $A$ and $C$ are independent, and $B$ and $C$ are independent, does it then hold that $A\cap B$ is independent of $C$ as well?

Answer 1

No.

Throw fair coins 1 and 2.

Let $A$ be the event of heads for coin 1 and let $B$ be the event of heads for coin 2.

Let $C$ be the event that both coins give the same result.

Then $A$ and $C$ are independent and also $B$ and $C$ are independent.

However, $A\cap B$ and $C$ are not independent.

Observe that $$P(A\cap B\cap C)=\frac14\neq\frac14\frac12= P(A\cap B)P(C)$$