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.
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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 using induction
Boole's inequality may be proved for finite collections of events using the method of induction.
For the
For the case
Since
Since
by the first axiom of probability, we have
and therefore
Proof without using induction
For any events in
One of the axioms of a probability space is that if
this is called countable additivity.
If
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 if
Therefore can we make following equation
Bonferroni inequalities
Boole's inequality may be generalised to find upper and lower bounds on the probability of finite unions of events. 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.