In the process of clinical trials and health-care evaluation, Bayesian approaches have increasingly become the center of attention. In this article, sample size calculations for a non-inferiority test of two independent binomial proportions in a clinical trial are considered in a Bayesian framework. The hybrid Neyman-Pearson-Bayesian (hNPB) probability, the conditionally Bayesian (cB) probability and the unconditionally Bayesian (uB) probability are formulated through a conjugate normal analysis. The sample sizes are calculated based on formulas where normal prior distributions are assumed, and are compared with the Neyman-Pearson (NP) sample size. Our results show that the sample size based on the hNPB probability allows us to critically evaluate the appropriateness of the NP sample size. It is suggested that the sample size calculated based on the cB probability formula is smaller than the NP sample size.