Boole's inequality

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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.

Contents

ProofEdit

Proof using inductionEdit

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

For the   case, it follows that

 

For the case  , we have

 

Since   and because the union operation is associative, we have

 

Since

 

by the first axiom of probability, we have

 ,

and therefore

 .

Proof without using inductionEdit

For any events in  in our probability space we have

 

One of the axioms of a probability space is that if   are disjoint subsets of the probability space then

 

this is called countable additivity.

If   then  

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 can we make following equation

 

Bonferroni inequalitiesEdit

Boole's inequality may be generalised 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.

See alsoEdit

ReferencesEdit

  1. ^ Casella, George; Berger, Roger L. (2002). Statistical Inference. Duxbury. pp. 11–13. ISBN 0-534-24312-6. 

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