Boole's inequality

In probability theory, Boole's inequality, named after George Boole, (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.

Formally, for a countable set of events "A"₁, "A"₂, "A"₃, ..., we have

:$$Prleft [igcup_{i} A_i ight] leq sum_i Prleft [A_i ight] .

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

Bonferroni inequalities

Boole's inequality may be generalised to find upper and lower bounds, known as Bonferroni inequalities, on the probability of finite unions of events.

Define::$$S_1 := sum_{i=1}^n Pr(A_i),::$$S_2 := sum_{iand for 2 < "k" ≤ "n",::$$S_k := sum Pr(A_{i_1}cap cdots cap A_{i_k} ),where the summation is taken over all "k"-tuples of distinct integers.

Then, for odd "k" ≥ 1,::$$Prleft( igcup_{i=1}^n A_i ight) leq sum_{j=1}^k (-1)^{j+1} S_j,and for even "k" ≥ 2,::$$Prleft( igcup_{i=1}^n A_i ight) geq sum_{j=1}^k (-1)^{j+1} S_j.

Boole's inequality is recovered by setting "k" = 1.

When "k" = "n", then equality holds and the resulting identity is the inclusion-exclusion principle.

References

ee also

* Inclusion-exclusion principle
* Carlo Emilio Bonferroni