Improving probability bounds by optimization over subsets

The simple device of maximization over subsets of events can provide substantial improvement over the Dawson–Sankoff 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 Kuai, Alajaji, and Takahara. In each of their examples, the maximized bound achieves the exact probability of the union using a subset of events containing no redundant events of the original set of events.