This dissertation is concerned with the estimation of three functions of the binomial probability of "success": estimation of a linear combination of independent binomial probabilities; fixed precision estimation of the binomial probability; and estimation of the logit transformation of the binomial probability. A Bayesian viewpoint is adopted temporarily to "suggest" a wide class of admissible estimators for each problem. Designated the class C of SBP estimators, it is the class of Bayes estimators derived from Symmetric Beta Priors (the class of conjugate priors for the binomial), and often includes the maximum likelihood estimator as a special case. Once a class of estimators for each problem is suggested by the Bayesian viewpoint, three criteria are used to obtain the "optimum" estimators in that class. Two of these criteria are classical in nature: minimax risk and minimax weighted risk. The third criterion, the solution of which is the estimator corresponding to the "least favorable" prior in the class of priors considered, is subjective in nature and would appeal more to Bayesians. For each of the three problems, a class C of SBP estimators is suggested, and the optimum estimators from this class are obtained. In addition, for a special case in the estimation of a linear combination of binomials, an estimator is found that is minimax among all estimators, as well as minimax among SBP estimators. / Ph. D.