A mastery test is frequently described as follows: an examinee responds to n dichotomously scored test items. Depending upon the examinee's observed (number correct) score, a mastery decision is made and the examinee is advanced to the next level of instruction. Otherwise, a nonmastery decision is made and the examinee is given remedial work. This document deals with the problem of determining an optimal passing score for a mastery test when the purpose of the test is to predict success or failure on an external criterion. For the case of constant losses for the two error types, a method of determining an optimal passing score is readily derived using standard techniques. The purpose of this research is to describe a lower bound to the probability of identifying an optimal passing score based on a random sample of examinees. The second section of this document deals with the necessity of estimating two parameters to approximate the maximum liklihood estimates using iterative technique. Using Monte Carlo techniques, the Newton-Raphson approximation is compared to other procedures. (Author/CE)