How to show $ \mathbb P(B \cap C) * \mathbb P(B\cup C)$?

Question

Let B,C be events in a probability space. Show $ \mathbb P(B\cup C) \mathbb P(B\cap C) \leq \mathbb P(B) \mathbb P(C) $.

My work: I started to use inclusion- exclusin principle on the left side, but without success. Could you give me a hint, please?

Answer 1

Let $r=P(A\cap B)$, $s=P(A\setminus B)$ and $t=P(B\setminus A)$. Your proposed inequality is then $$r(r+s+t)\le(r+s)(r+t).$$ That does simplify rather nicely...