This note has given results for the probability of committing a Type I error in first-order stochastic dominance when it is assumed that both options have been sampled from the same distribution of returns. The probabilities are based on known results in the stochastic processes and statistical goodness-of-fit literatures, but have not been recognized in connection with the problem of estimation of risk in stochastic dominance. Further work on FSD Type II error probabilities based on ideas in this paper is in progress.