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Boole's inequality

In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. Boole's inequality is named after George Boole.

Formally, for a countable set of events A1, A2, A3, ..., we have

In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive.

Proof

Proof using induction

Boole's inequality may be proved for finite collections of events using the method of induction.

For thecase, it follows that
For the case, we have
Sinceand because the union operation isassociative, we have

Since

by the first axiom of probability, we have

and therefore

Proof without using induction

For any events inin ourprobability spacewe have
One of the axioms of a probability space is that ifare disjoint subsets of the probability space then

Ifthen

Indeed, from the axioms of a probability distribution,

Note that both terms on the right are nonnegative.

Now we have to modify the sets, so they become disjoint.
So if, then we know

Therefore, we can deduce the following equation

Bonferroni inequalities

Boole's inequality may be generalized to find upper and lower bounds on the probability of finite unions of events.[1] These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni, see Bonferroni (1936).

Define

and

as well as

for all integers k in {3, ..., n}.

Then, for odd k in {1, ..., n},

and for even k in {2, ..., n},

Boole's inequality is recovered by setting k = 1. When k = n, then equality holds and the resulting identity is the inclusion–exclusion principle.

• Diluted inclusion–exclusion principle

• Schuette–Nesbitt formula

• Boole–Fréchet inequalities

References

[1]
Citation Linkbooks.google.comCasella, George; Berger, Roger L. (2002). Statistical Inference. Duxbury. pp. 11–13. ISBN 0-534-24312-6.
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