If A$,B$ and $C$ are random events in a sample space and if $A, B$ and $C$ are pairwise independent and $A$ is independent of $(B \cup C)$, then is it true that $A,B$ and $C$ are mutually independent.
My Attempt : (with questions of this type at my college the answer is usually in the affirmative.)
So to prove that $A,B$ and $C$ are mutually independent, all that remains to show is that $P(A \cap B \cap C) = P(A) \times P(B) \times P(C)$. We know that $P(A \cap (B \cup C))=P(A) \times P(B \cup C)$. How do I go about this?