Suppose that:
the events $A$ and $B\cap C$ are independent.
the events $B$ and $A\cap C$ are independent.
the events $C$ and $A\cap B$ are independent.
the events $A$ and $B\cup C$ are independent.
$P(A),P(B),P(C)$ are nonzero.
Prove that $A,B,C$ are mutually independent.
I've noticed that $P(A\cap B \cap C)=P(A)P(B\cap C)=P(B)P(A\cap C)=P(C)P(A\cap B)$.
What then? I'm new to probability, and I may be missing some standard trick…
Best Answer
$ P(A \cap (B\cup C)) = P(A \cap B)+ P(A \cap C) - P(A \cap B \cap C)$
$P(A \cap (B\cup C)) = P(A ) P(B\cup C))= P(A )(P(B)+P(C)-P(B \cap C))$
$\Rightarrow P(A \cap B)+ P(A \cap C)= P(A )(P(B)+P(C))$
$ \Rightarrow P(B)P(C)( P(A \cap B)+ P(A \cap C))= P(A ) P(B)P(C) (P(B)+P(C)) $
$ \Rightarrow P(A \cap B\cap C)= P(A ) P(B)P(C) $