Occasionally, situations arise where mixtures of two binomials with one known success parameter are met. An example in educational testing is the mastery or random guessing model in which an examinee is supposed either to master the items or not to master them and to guess blindly. This paper gives moment estimators for such mixtures and presents results from a Monte Carlo investigation into their statistical properties. The results suggest excellent estimators that can safely be used in most instances. It also indicates how the properties of these estimators relate to those of moment estimators for the case in which both success parameters are unknown. Finally, it is pointed out that in situations in which errors in specifying the true value of the known parameter may occur, it might be prudent to consider this parameter as unknown and to estimate accordingly. (Four data tables are included.) (Author)