To answer whether patients' race belongs in clinical prediction algorithms, two types of prediction models are considered: (i) diagnostic, which describes a patient's clinical characteristics, and (ii) prognostic, which forecasts a clinical risk or treatment effect that a patient is likely to experience in the future. The ex ante equality of opportunity framework is used, where specific health outcomes, which are prediction targets, evolve dynamically due to the effects of legacy levels of outcomes, circumstances, and current individual efforts. In practical settings, this study shows that failure to include race corrections will propagate systemic inequities and discrimination in any diagnostic model and specific prognostic models that inform decisions by invoking an ex ante compensation principle. In contrast, including race in prognostic models that inform resource allocations following an ex ante reward principle can compromise the equality of opportunities for patients from different races. Simulation results demonstrate these arguments.