Boole's Inequality - 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.