# Boole's Inequality

From ProofWiki

## Theorem

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $A_1, A_2, \ldots, A_n$ be events in $\Sigma$.

Then:

- $\displaystyle \Pr \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n \Pr \left({A_i}\right)$

## Proof

A direct consequence of the facts that:

- The result Finite Union of Sets in Subadditive Function which gives:

- $\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$

for a subadditive function $f$.

$\blacksquare$

## Also known as

This inequality is also known as **Union Bound**.

## Source of Name

This entry was named for George Boole.

## Sources

- Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*(1986)... (previous)... (next): $\S 1.11$: Problems: $3$