A systematic method that quantitatively assesses property prediction uncertainty (imprecision) on optimal molecular design problems is introduced. Property - structure relations are described with specific nonlinear functionalities based on group contribution methods. Property prediction uncertainty is explicitly quantified by using multivariate probability density distributions to model the likelihood of different realizations of the group contribution parameters. Assuming stability of these probability distributions, a novel approach is introduced for transforming the original nonlinear stochastic formulation into a deterministic MINLP problem with linear binary and convex continuous parts with separability. The resulting convex MINLP formulation is solved to global optimality for molecular design problems involving many uncertain group contribution parameters. Results indicate the computational tractability of the method and the profound effect that property prediction uncertainty may have in optimal molecular design. Specifically, trade-off curves between performance objectives, probabilities of meeting the objectives, and chances of satisfying design specifications offer a concise and systematic way to guide optimal molecular design in the face of property prediction uncertainty.