Addition Theorem of Probability

The addition theorem gives the probability that at least one of two events occurs — the probability of the union $A \cup B$.

The subtraction of $P(A \cap B)$ is essential: outcomes common to both $A$ and $B$ are counted once in $P(A)$ and again in $P(B)$, so they would be double-counted unless we remove them once. (This is exactly the inclusion–exclusion idea for the sizes of sets.)

More generally, for mutually exclusive events $A_1, A_2, \ldots, A_k$,

Example. Draw one card from a pack. Let $A$ = "the card is a heart" and $B$ = "the card is a king". Then $P(A) = \dfrac{13}{52}$, $P(B) = \dfrac{4}{52}$, and $A \cap B$ = "the king of hearts", a single card, so $P(A \cap B) = \dfrac{1}{52}$. The probability the card is a heart or a king is …