Linear programming bounds on the union probability

Lower and upper bounds on the union probability for N events are derived in terms of the individual and pairwise event probabilities by solving a linear program with (Formula presented.) variables. The bounds, which can be efficiently determined, are shown to be optimal when (Formula presented.) and are always sharper than recent optimal bounds which use slightly less information. Their competitive sharpness is also illustrated via numerical comparisons with state-of-the-art bounds in the literature.