We consider a situation in which a central authority must allocate non-tradeable and non-marketable goods between a group of individuals in a fair way. There are exogenous divisibility constraints imposed on the goods to be allocated. The authority has absolutely no information on the preferences of the recipients; moreover, no interaction is allowed among recipients or between the authority and the recipients. Envy-freeness is the equity criterion adopted. Using a remarkable property of simplices (which we introduce and prove) we argue that assigning bundles of equal expected value (forming what is called in this paper the class of “balanced allocations”) is hardly fair unless extra effort is made to discriminate between these proposed allocations.