Improving probability bounds by optimization over subsets (Q2368926)

The simple device of maximization over subsets of events can provide substantial improvement over the \textit{D. A. Dawson} and \textit{D. Sankoff} [Proc. Am. Math. Soc. 18, 504--507 (1967; Zbl 0154.19401)] degree two lower bound on the probability of a union of events and can also exceed a sharper bound that uses individual and pairwise joint event probabilities developed by \textit{H. Kuai, F. Alajaji} and \textit{G. Takahara} [Discrete Math. 215, No. 1--3, 147--158 (2000; Zbl 0960.60017)]. In each of their examples, the maximized bound acquires the exact probability of the union using a subset of events containing no redudant events of the original set of events.