# Boole's Inequality

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## Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Then:

- $\displaystyle \map \Pr {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map \Pr {A_i}$

## Proof

A direct consequence of the facts that:

- the result Finite Union of Sets in Subadditive Function which gives:
- $\displaystyle \map f {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map f {A_i}$

- 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

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