Boole's inequality

The Bonferroni inequalities are formulas used to estimate the probability of the average or the union of events.

Designation according to Bonferroni

The Bonferroni inequalities are not necessarily named after Carlo Emilio Bonferroni cope.

Bonferroni probably was not the originator of these inequalities, but they used it to define a statistical estimator ( Bonferroni method). The naming after him is therefore particularly popular in statistical circles. Due to its simplicity, the inequalities likely have been known before him.

The first of the following inequalities is more commonly referred to George Boole as Boolean inequality; often the inequalities are also called no name reference.

First inequality

Below are arbitrary events in a probability space. Denote the probability of the event and the union of the events. Then:

It also applies more generally:

These inequalities are also called Boolean inequalities.

Evidence

Substituting

Then the pairwise disjoint and it is

This follows

Here, the second equality holds because of the σ - additivity and because of the inequality and the monotonicity of the probability measure.

Second inequality

Below are arbitrary events in a probability space again. Furthermore, the complement of call. Then follows:

Examples

• Be the set of results of a dice roll. Denote the event of rolling an even number and the event, at least to roll a 5. Obviously applies and. After the first Bonferroni inequality holds for the event, an even number, or at least roll a 5

• The scenario is as in the previous example. After the second Bonferroni inequality holds for the event, an even number and dice at least a 5