---
tag: 0P0MZ4X682
kind: definition
name: Independence of two events
msc: [60A05]
defines: 03AHF0XRVG
uses: [0ZYBZ34RT2, 00SQGJNRTB, 0FJB06AD2P]
sources:
  - url: https://en.wikipedia.org/wiki/Independence_(probability_theory)
    locator: Two events
---

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
The events $A$ and $B$ are independent if
$$P(A \cap B) = P(A) \, P(B).$$

## Remarks

If $P(B) > 0$, then $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$, with the [[conditional probability|0FJB06AD2P]].
If $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$.
An event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \in \{0, 1\}$.
Independence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ are never independent.

## Explanations

### L1 human

Two events are independent when knowing that one happened tells you nothing about the other.
Roll a fair die two times. The events "the first roll is $6$" and "the second roll is $6$" are independent.
Both happen with probability $\frac16 \cdot \frac16 = \frac{1}{36}$.