Bounds on the Probability of a Finite Union
With an Application to Modulation Error Analysis

Let A[1],...,A[N] be N arbitrary events. The inclusion-exclusion formula for the probability of their union taught in any introductory probability course is well-known; however it involves knowing the probability of intersections of 3 and more events. In many applications the probabilities of these higher order intersections are extremely difficult or impossible to evaluate, and it is of interest to obtain bounds that depend at most on pairwise intersection probabilities. The first term in the inclusion-exclusion formula gives the union upper bound: P(A[1],..,A[n])<=P(A[1])+..+P(A[n]); the first two terms in the formula give a lower bound. In this talk some recent upper and lower bounds are discussed - these include an extension of a linear programming bound discussed by Dawson and Sankoff in 1968, and two algorithmic bounds, one an upper bound based on a minimal spanning tree and one a lower bound based on the stepwise search algorithm. Performance of these bounds is illustrated in estimating the symbol and bit error probabilities of modulation schemes for a digital communications channel.