Bounding the Probability of the Union of Events by the Use of Aggregation and Disaggregation in Linear Programs Bounding the Probability of the Union of Events by the Use of Aggregation and Disaggregation in Linear Programs

Given a sequence of n arbitrary events in a probability space we as sume that the individual probabilities as well as some or all joint probabilities of up to m events are known where m n Using this information we give lower and upper bounds for the probability of the union The bounds are obtained as optimum values of linear programming problems or objective function values corre sponding to feasible solutions of the dual problems If all joint probabilities of the k tuples of events are known for k not exceeding m then the LP is the large scale Boolean probability bounding problem Another type of LP is the binomial moment problem where we assume the the knowledge of some of the binomial moments of the number of events which occur The two LP s can be obtained from each other by aggregation disaggregation procedures In this paper we de ne LP s which are obtained by partial aggregation disaggregation from these two LP s This way we can keep the size of the problem low but can produce very good bounds in many cases One of our new results is a generalization of Hunter s upper bound for the probability of the union of events Numerical examples are presented