if probability of $A$ and $C$ are independent and probability of $B$ and $C$ are independent, does it imply $A \cap B \cap C$ are independent?

Question

I know that the answer is it's not always the case that $A \cap B \cap C $ holds.

I'm not sure why. I tried to do this mathematically: Because A, B, C are independent: if $P(A \cap C) \cap P(B \cap C)$ then $ (P(A) * P(C)) * (P(B) * P(C))$ $\neq$ $ P(A) * P(B) * P(C) $

I know this is not accurate. Any way to show this more succinctly is appreciated. Thanks

Answer 1

First, recall that it is not “probability of two events are independent”, but “two events are independent”.

Second, probabilities are numbers so you can’t compute an intersection of probabilities. It just does not make sense.

For intuition, think that among three events, any two can be completely uncorrelated but the three may have some redundancy. A very general example is if $A$ and $B$ are independent and have probability $1/2$, then $C=A\Delta B$ (“either $A$ and $B$ occur, either both don’t occur”) is independent from $A$ and independent from $B$. However, if you know the results of $A$ and $B$, you know also if $C$ occurred, thus $A,B,C$ can’t be independent.