TITLE: Why is $P(a \text{ and } b)$ maximized when $P(a \text{ or } b)$ is minimized?
QUESTION [6 upvotes]: I can't seem to wrap my head around why $P(a \text{ and } b)$ is minimized when $P(a \text{ or } b)$ is maximized. This comes from PIE:
$$P(a \text{ or } b) = P(A) + P(B) - P(a \text{ and } b).$$
Can someone please explain the intuition behind this? I'm even trying to picture the Venn diagram in my head, but this exact relationship doesn't make sense.

REPLY [9 votes]: $A \cap B$ is the overlapping part in $A \cup B$.
Imagine you have two sheets of paper and you want to maximize their surface.
If they overlap, you can increase the surface by making them overlap less.