Uniformity, that is, equiprobability of all available options is central as a theoretical presupposition and as a computational tool in probability theory. It is justified only when applied to an appropriate sample space. In five studies, we posed diversified problems that called for "unequal" probabilities or weights to be assigned to the given units. The predominant response was choice of "equal" probabilities and weights. Many participants failed the task of partitioning the possibilities into elements that justify uniformity. The uniformity fallacy proved compelling and robust across varied content areas, tasks, and cases in which the correct weights should either have been directly or inversely proportional to their respective values. Debiasing measures included presenting individualized and visual data and asking for extreme comparisons. The preference of uniformity obtains across several contexts. It seems to serve as an anchor also in mathematical and social judgments. People's pervasive partiality for uniformity is explained as a quest for fairness and symmetry, and possibly in terms of expediency. (Contains 6 tables and 1 figure.)