Bonferroni Inequalities

Theorem
The Bonferroni inequalities are a generalization of Boole's Inequality:

Let $E = \set {E_1, E_2, \ldots, E_n}$ be a set of $n$ events.

Let $\overline E = \set {\overline {E_1}, \overline {E_2}, \ldots, \overline {E_n} }$ be the set of complementary events to each of $\set {E_1, E_2, \ldots, E_n}$ respectively.

Let:

where:
 * $\map \Pr {E_i}$ denotes the probability of $E_i$
 * $\bigcap E$ denotes the intersection of $E$.

When $n$ is odd:

and:

When $n$ is even:

and:

In particular:
 * $\map \Pr {\bigcap E} > 1 - \ds \sum_{i \mathop = 1}^n \map \Pr {\overline E_i}$

which is a statement of Boole's Inequality.

Also see

 * Bonferroni Correction