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The need to reason with imprecise probabilities arises in a wealth of situations ranging from pooling of knowledge from multiple experts to abstractiondbased probabilistic planning. Researchers have typically represented imprecise probabilities using intervals and have developed a wide array of different techniques to suit their particular requirements. In this paper we provide an analysis of some of the central issues in representing and reasoning with interval probabilities. At the focus of our analysis is the probability crossdproduct operator and its interval generalization, the ccdoperator. We perform an extensive study of these operators relative to manipulation of sets of probability distributions. This study provides insight into the sources of the strengths and weaknesses of various approaches to handling probability intervals. We demonstrate the application of our results to the problems of inference in interval Bayesian networks and projection and evaluation of abstract probabilistic plans.