This paper considers stochastic social choice rules which, for every feasible set of alternatives and every profile of individual orderings, specify social choice probabilities for the feasible alternatives. It is shown that if such a stochastic social choice rule satisfies (1900): (i) a probabilistic counterpart of Arrow's independence of irrelevant alternatives, (ii) ex-post Pareto optimality, and (iii) "regularity" (a "rationality" property postulating that given the individual preference orderings, if the feasible set of alternatives is expanded, then the social choice probability for an initially feasible alternative cannot increase), then the power structure under it is almost completely characterized by weighted random dictatorship.