Lower bounds on the probability of a finite union of events

In this paper, lower bounds on the probability of a finite union of events are considered, i.e., P(U^N_i=1A_i, in terms of the individual event probabilities {P(Ai), i = 1, ... ,N} and the sums of the pairwise event probabilities, i.e., {Σ_j:j≠i P(Ai ∩ Aj), i = 1, ... ,N}. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the P(Ai)'s and σj:j≠ij P(Ai ≠ Aj)'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with N² - N + 1 variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, Alajaji, and Takahara [Discrete Math., 215 (2000), pp. 147-158]; (iii) numerical examples are provided to illustrate the performance of the bounds.