"Combination of events" made easy

Abstract

The probability that precisely r out of n overlapping events shall occur is $S_r - \left( {\begin{array}{*{20}c}{r + 1} \\r \\\end{array} } \right)S_{r + 1} + \left( {\begin{array}{*{20}c}{r + 2} \\r \\\end{array} } \right)S_{r + 2} - \ldots \pm \left( {\begin{array}{*{20}c}n \\r \\\end{array} } \right)S_n$. This result and the related result for the ‘tail’ probability that more than r shall occur are quite standard.

Here we give a proof of this group of results which could in some cases have advantages over the currently accepted proofs. Our proofs make no explicit use of special identities relating to binomial coefficients. We go on to consider events of two kinds — type E and type F — and establish the similar results for the probability that precisely r of type E and precisely s of type F shall happen, and for the corresponding tail probabilities.

A proof in the same spirit — no technical knowledge of binomial identities — is given for Bonferroni’s inequalities.