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In comparing two treatments with dichotomous responses, the randomized play-the-winner rule (Wei and Durham 1978) tends to assign more study subjects to the better treatment. For ethical reasons, this property is desirable for studies on human subjects. The randomized play-the-winner rule, which is a modification of Zelen's play-the-winner rule (Zelen 1969), is not deterministic and is less vulnerable to experimental bias than other adaptive designs. Recently, this design has been used in a trial to evaluate extracorporeal membrane oxygenation (ECMO) for treating newborns with respiratory failures at the University of Michigan. In this article, exact conditional, exact unconditional, and approximate confidence intervals for the treatment difference are studied from a frequentist point of view with the randomized play-the-winner rule. For small and moderate-sized trials, the exact unconditional procedures perform …