# Boole's inequality

## Inequality applying to probability spaces / From Wikipedia, the free encyclopedia

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Boole's inequality?

Summarize this article for a 10 year old

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. This inequality provides an upper bound on the probability of occurrence of at least one of a countable number of events in terms of the individual probabilities of the events. Boole's inequality is named for its discoverer, George Boole.^{[1]}

This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2012) |

Formally, for a countable set of events *A*_{1}, *A*_{2}, *A*_{3}, ..., we have

- ${\mathbb {P} }\left(\bigcup _{i=1}^{\infty }A_{i}\right)\leq \sum _{i=1}^{\infty }{\mathbb {P} }(A_{i}).$

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