This paper introduces a probabilistic model that is capable of diagnosing and classifying cognitive errors in a general problem-solving domain. Item response theory is used to deal with the variability of response errors. Responses from a 38 item fraction addition test given to 595 junior high school students are used to illustrate the model. Through error analysis the students' responses are classified into 34 groups of misconceptions, and validated with the error diagnostic program written by a typical formal logic approach. The model can control classification of test performance by selecting a particular set of ellipses. These ellipses represent erroneous rules in a procedure domain, or sources of misconceptions producing a variety of erroneous rules of operation, or different achievement levels. Applications of the model include: (1) computer aided instruction on microcomputers as an integrated part of training programs; and use as a placement tool by training program administrators. (LMO)