Mutual Independence implies Pairwise Independence

Question

How to prove that mutual independence of events implies pairwise independence?

That is, prove that if $P(A \cap B \cap C) = P(A)P(B)P(C)$ then A, B are independent. B, C are independent. C, A are independent.

I tried using the sum rule to prove but couldn't separate two events from the third.

Answer 1

Let prove that $A$ and $B$ are independent. I let you adapt for $A$ and $C$. \begin{align*} \mathbb P(A\cap B)&=\mathbb P\big((A\cap B\cap C)\cup (A\cap B\cap C^c)\big)\\ &=\mathbb P(A\cap B\cap C)+\mathbb P(A\cap B\cap C^c)\\ &=\mathbb P(A)\mathbb P(B)\mathbb P(C)+\mathbb P(A)\mathbb P(B)\mathbb P(C^c)\\ &=\mathbb P(A)\mathbb P(B)\big(\mathbb P(C)+\mathbb P(C^c)\big)\\ &=\mathbb P(A)\mathbb P(B). \end{align*} I let you justify each step.