Mutual independence of three events

Question

Is it possible to have three events $A,B,C$ such that $A$ is mutually independent to $B,C$ and $B$ is NOT mutually independent to $A,C$.

By mutual independence I mean, $A$ is mutually independent to a set $S$ if $$P\left[A\left|\bigcap_{X\in S} X\right.\right] = P[A]$$ for all subsets $X$ of $S$.

I tried thinking of it in terms of rolling two dice. Having $A,B$ be the events such that the first die results in something, and the second die results in something else. Also having $C$ be the sum of the two dice resulting in some value. But no matter what events I choose for $A,B,C$ I can't come up with ones that fit my constraints. Any help would be appreciated.

Answer 1

$A$ being mutually independent of $B$ means $P(A\cap B) / P(B) = P(A)$.

That means $P(A\cap B) = P(B) P(A)$, so if $A$ is mutually independent of $B$, then $B$ is mutually independent of $A$. You can make $A$, $B$ two rolls of two distinct dice, and $C$ to be the exact same event as $B$. Then $A$ is independent to $B,C$, and $B$ is independent to $A$, but $B$ is not independent to $C$, because $B$ is $C$.

Answer 2

Yes, this is possible. One example would be the case that $B,C$ refer to the same event, and $A$ is independent of $B$. Then $A$ is mutually independent to $B,C$ but $B$ is not mutually independent to $A,C$.

You can easily construct cases in which $B,C$ are not identical events. E.g., $A$ is the event that die 1 comes up even, $B$ is the event that die 2 comes up odd, $C$ is the event that die 2 comes up prime.