This paper explores the possibilities opened up by combining preference aggregation and randomization in passing from individual to collective judgements about alternatives. We study the distribution of power under functions which assign to each profile of individual preferences a probabilistic judgement on each pair of alternatives x, y, that is, a probability of x being considered socially at least as good as y. Conditions are described under which the power of coalitions to guarantee that an alternative is not declared worse than another is subadditive, while their power to guarantee that one alternative is not declared better than another is superadditive. These results extend some of the main findings on the structure of decisive coalitions under deterministic social choice rules.