Using the concept of selectors of random sets, we provide an interpretation for numerical degrees of possibility. The axioms (and hence the calculus of possibilities) of possibility measures are justified, in the context of random sets, on the basis that possibility distributions, as covering functions, lead to maxitive capacity functionals of random closed sets. Also, possibility measures appear as limits of probability measures in the study of large deviations principle, and as such, the idempotent operator max is justified. The problem of admissibility of possibility measures is also discussed.