Addition rule for two events

Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ be events. Then $$P(A \cup B) + P(A \cap B) = P(A) + P(B).$$ Equivalently, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Remarks

If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$. The rule is the probabilistic form of inclusion–exclusion for two sets. The inclusion–exclusion principle gives the version for $n$ events. A consequence is $P(A \cup B) \le P(A) + P(B)$, the case $n = 2$ of Boole's inequality.

Explanations

L1 human

When you add $P(A)$ and $P(B)$, you count the outcomes in both events twice, so you subtract $P(A \cap B)$ once. For one card from a deck, $P(\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52$.