Often, in dose-response studies, it is desirable to arrange treatments in a region defined by the unknown surface. We accomplish such arrangements using sequential designs that are functions of previous outcomes. Here, for a class of randomized up-and-down designs for dichotomous responses (e.g., toxicity/no toxicity) and predetermined sample size, distributional results for the total number of toxic responses are derived. These results are useful in selecting sequential design parameters and have obvious ethical implications.