An extension of the method of polynomials and a new reduction formula for Bonferroni-type inequalities

We prove that Bonferroni-type inequalities on the probability of at least r events occurring out of n are valid if, and only if, they are valid for a triangular array of independent events. Such method for proof was so far available for the case of exactly r events occurring. This new method allows us to reduce the mentioned Bonferroni-type inequalities to the special case of none occurring. This reduction method is exploited to establish a large class of new inequalities