Starting with two interpretations of probability, viz. relative frequencies of attributes in infinite sequences, and betting quotients under weak coherence, the author establishes upper and lower probabilities [à la \textit{I. J. Good's} paper, Logic, Methodology and Philosophy of Science, Proc. 1960 internat. Congr. 319-329 (1962; Zbl 0192.021)]. This leads in turn to the adoption of an orthomodular partially ordered set as the domain of definition of a point-valued probability distribution. The main aim of the paper is to investigate the relevance of this approach to quantum mechanics, and it is shown that ``there can be no phase-space representation of the vast majority of quantum systems employing upper and lower probabilities''. The author concludes by proposing a generalized conception of probability that permits both objective and subjective interpretations of quantum mechanics.