Improved Bounds on the Probability of a Union and on the Number of Events that Occur
Ilan Adler, Richard M. Karp, Sheldon M. Ross
TL;DR
The paper addresses bounds on $P(X \ge r)$ where $X$ counts how many of the $n$ events $A_1,\dots,A_n$ occur, given the probabilities of all $s$-fold intersections up to size $k+1$. It develops a combinatorial–algebraic framework that yields representations of the distribution in terms of the sums $S_j$ and higher-order moments $E[{X-i \choose k-i+1}]$, enabling stronger, more information-rich bounds than classical Bonferroni-type inequalities. A key contribution is a parity-based family of bounds in terms of $S_j$ (and $S_{k+1}$) plus correction terms, plus a refinement that expresses these corrections via actual $s$-fold intersection probabilities. An especially notable result is an improved bound for $P(X \ge 1)$ obtained via a random-permutation argument, introducing a max-term construction over subsets to tighten the bound. These results sharpen traditional inequalities and provide practical, computable bounds when higher-order intersection data is available, with potential impact in risk assessment and reliability analyses of dependent events.
Abstract
Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question dates back to Boole in the 19th century, and it is well known that the odd partial sums of the Inclusion- Exclusion formula provide upper bounds, while the even partial sums provide lower bounds. We give a combinatorial characterization of the error in these bounds and use it to derive a very simple proof of the strongest possible bounds of a certain form, as well as a couple of improved bounds. The new bounds use more information than the classical Bonferroni-type inequalities, and are often sharper.