Non-orthogonal multiple access (NOMA) relies on power domain multiplexing via successive interference cancellation (SIC). Current NOMA system designs are based on information rate and assume perfect SIC. However, error propagation in SIC is inevitable in practice. This paper considers imperfect SIC and investigates the NOMA system design from the perspective of minimum error probability. Specifically, we consider the uncoded and the coded NOMA systems and derive the error probabilities of the users. Then, we aim to minimize the average error probability of the users by searching for appropriate power allocation. Numerical results are provided to evaluate the minimum error performance of NOMA along with useful insights.