Let A1,A2,…,An be events on a given probability space, and let mn be the number of those Aj which occur. Put S0 = S0,n = 1, and
where the summation is over all subscripts satisfying 1 ≤ i1 < i2 < … < ik ≤ n. For convenience in some formulae we adopt the convention Sk, n = 0 if k > n. By turning to indicator variables one immediately finds that
Inequalities of the form
where ck = ck(r, n) and dk = dk(r, n) are constants (not dependent on the events Aj, 1 ≤ j ≤ n), possibly zero, are called Bonferroni-type inequalities. This same name applies if P(mn = r) is replaced by P(mn ≥ r) in the middle. The best known such inequalities are the method of inclusion and exclusion
where j ≥ 0 is an arbitrary integer. An extension of (4) to arbitrary r, called Jordan's inequalities (see Takács [16]), is as follows: for 0 ≤ r ≤ n and for any integer j ≥ 0,
It is observed by Galambos and Mucci[9] that (5) follows from (4), and indeed, one can always generate inequalities of the form (3) for arbitrary r from the special case r = 0 if one utilizes only Sr,Sr+1… in the case of P(mn = r). As a matter of fact, from the instructions of Galambos and Mucci one has that the inequalities
hold for an arbitrary sequence A1,A2,…,An of events if, and only if, for an arbitrary sequence A1,A2,…,An−r of events,
where