On the classical Bonferroni inequalities and the corresponding Galambos inequalities

Abstract

Let (A 1 A 2, · ··, An ) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds. Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.