Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow for an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class TμMP of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing TμMP are compatible with classical mechanics (CM), statistical mechanics (SM), and QM, hence we assume that these theories belong to TμMP. In the case of CM and SM, this assumption is irrelevant, as all of the notions introduced in them as members of TμMP reduce to standard notions. In the case of QM, it leads to interpret quantum probability as a derived notion in a Kolmogorovian framework, explains why it is non-Kolmogorovian, and provides it with an epistemic interpretation. These results were anticipated in a previous paper, but they are obtained here in a general framework without referring to individual objects, which shows that they hold, even if only a minimal (statistical) interpretation of QM is adopted in order to avoid the problems following from the standard quantum theory of measurement.