Algorithms based on probability theory can address issues of uncertainty directly through their representational framework and their theory for data combination. In this paper, we discuss the advantages of probabilistic formulations for molecular-structure calculations, describe one implementation of such a formulation, and show its performance on a data set derived from analysis of the statistical correlations within a set of aligned transfer RNA sequences. By assigning reasonable physical interpretations to certain statistical correlations, we are able to calculate three-dimensional structures for tRNA from a random starting structure. The constraints that we use are associated with different variances, and so their effects are not uniform, and must be reconciled by a probabilistic algorithm to yield the most likely structure. As might be predicted, the uncertainty in the position for each base is a function of both the number and strength of the constraints, and is reflected in the variances in atomic position calculated by the algorithm. For example, the hinge region in the tRNA is shown to be the most uncertain. In addition, the algorithm retains information about positional covariation that is useful for understanding the relationships between different parts of the structure. These experiments also demonstrate that we can define a single-sphere representation for each base that is useful for nucleic acid structural calculations in the same way that alpha-carbon representations are useful for protein structural calculations.