Boole's Inequality

A Boole's Inequality is an inequality relation that states the total probability of a finite or countable set union is no greater than the sum of the individual probabilities.

• AKA: Union Bound.
• Context:
  • It can be expressed as [math]\displaystyle{ P(\bigcup_{i} E_i) \le \sum_i P(E_i) }[/math], where P is the probability function,[math]\displaystyle{ E_i }[/math] is i-th event, and [math]\displaystyle{ \bigcup_{i} }[/math] is the union set operator.
• See: Bonferroni Inequality, Kolmogorov's Inequality, Markov's Inequality.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/ Retrieved 2016-08-28
  • 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

[math]\displaystyle{ {\mathbb P}\biggl(\bigcup_{i} A_i\biggr) \le \sum_i {\mathbb P}(A_i). }[/math]

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

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