---
tag: 0EGSEE9J9R
kind: statement
role: theorem
name: Boole's inequality
aliases: [union-bound]
msc: ["60E15"]
uses: [0ZYBZ34RT2, 00SQGJNRTB]
sources:
  - url: https://en.wikipedia.org/wiki/Boole's_inequality
    locator: lead section
same_as:
  wikidata: Q517378
  wikipedia: Boole's inequality
---

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A_1, A_2, A_3, \dots$ be a finite or infinite sequence of [[events|00SQGJNRTB]].
Then
$$P\left(\bigcup_{i} A_i\right) \le \sum_{i} P(A_i).$$

## Remarks

The inequality is also called the union bound. It needs no hypothesis on the events: they need not be disjoint or independent.
Equality holds when the events are pairwise disjoint.
The Bonferroni inequalities refine it with the terms of the [[inclusion–exclusion principle|0FF7726C7P]]. For example, $P\left(\bigcup_{i=1}^n A_i\right) \ge \sum_i P(A_i) - \sum_{i < j} P(A_i \cap A_j)$.

## History

The inequality is named after George Boole.

## Explanations

### L1 human

The chance that at least one of several bad things happens is at most the sum of their chances.
If each of $10$ parts fails with probability $0.01$, then some part fails with probability at most $0.1$.