---
tag: 0BZQ6ZX584
kind: statement
role: proposition
name: Addition rule for two events
aliases: [inclusion-exclusion-for-two-events]
msc: [60A05]
uses: [0ZYBZ34RT2, 00SQGJNRTB, 0C4YEQ9H5E]
sources:
  - url: https://en.wikipedia.org/wiki/Probability_axioms
    locator: Elementary consequences
---

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
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|0C4YEQ9H5E]]. The [[inclusion–exclusion principle|0FF7726C7P]] 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|0EGSEE9J9R]].

## 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, $P(\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52$.