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:

a Probability Measure is Subadditive
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