In a previous paper, the author proposed methods to obtain sharp lower and upper bounds for probabilities that at least one out of n events occurs, based on the knowledge of some of the binomial moments of the number of events which occur and linear programming formulations. This paper presents further results concerning other and more general logical functions of events: It gives sharp lower and upper bounds for the probabilities that: (a) exactly r events, (b) at least r events occur, using linear programming. The basic facts are expressed by the dual feasible basis characterization theorems which are interpreted in terms of the vertices of the dual problems. The paper mentions some linear inequalities, among the binomial moments, generalizes the theory for the case of nonconsecutive binomial moments and presents numerical examples.
