Abstract: Under the criterion of the set inclusion, two-sided admissible 1 - α confidence intervals for the probability of success for a binomial random variable are constructed using a new iterative method that is based on a direct analysis of coverage probability. A refined Clopper-Pearson interval is derived and compared with the Blyth-Still-Casella interval, and is recommended for statistical practice due to its performance and accessibility. A generalization is provided to the case of a discrete sample space with a single parameter distribution. Some details and an R-code that computes the refined Clopper-Pearson interval are given in Supplementary Materials.

Key words and phrases: Binomial distribution, coverage probability, set inclusion.