Two events, and the word or. Write $A \cup B$ for A or B, or both — the cup-shaped symbol is the union, and it is generous: it accepts anything that satisfies either event. Its partner $A \cap B$ is the intersection, read A and B, the outcomes where both hold at once.

If the two events cannot possibly happen together — mutually exclusive — the shares simply add: $P(A \cup B) = P(A) + P(B)$. If they CAN overlap, that sum charges the overlap twice, once inside A and once inside B, so it gets refunded exactly once: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, read aloud P of A-union-B equals P of A plus P of B minus P of A-intersect-B. Draw two overlapping circles; the lens in the middle is the whole lesson.

The exclusive rule is not a different law, it is this one with a zero in it. And the relation has four slots, so it runs backwards too: hand it any three and it returns the fourth — which is how a question about an unknown OVERLAP is answered from a known union. Keep the range check armed throughout. A union that comes out above 1 means the overlap never came off.