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 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 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 induction

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, 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. These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni; see.
Define
and
as well as
for all integers k in.
Then, for odd k in,
and for even k in,
Boole's inequality is the initial case, k = 1. When k = n, then equality holds and the resulting identity is the inclusion–exclusion principle.