Analysis of dependent probability systems of a moderate to large finite number (m) of events must often be based solely on the partial system information given byS1, the sum of the probabilities of occurrence of themindividuaf events; andS2, the sum of the probabilities of occurrence of each of the (m2) pairs of events. This information is used to develop the most stringent upper and lower bounds on the aggregated probabilitiesP[n], the probability of occurrence of exactlynevents (n= 0, 1, …,m); and onP1, the probability of occurrence of one or more of themevents. The most stringent bounds onP[m]andP1are more stringent than corresponding bounds presented in the literature, for most systems.