Retake Opportunities, Pass Probabilities, and Preparation for Exams

Additional retake opportunities generally increase the probability of eventually passing a given threshold at given competence, and decrease preparation for exams. Preparation work performed before the first attempt may increase only for very weak students, and may decline so much as to decrease the total pass probability only for very strong students. If additional preparation is possible before retaking a failed exam, it is optimal for students to make early attempts without much preparation. Some will be lucky enough to pass, and others will make many attempts after gradually improving preparation. Hence, plentiful retake opportunities reduce the reliability of exam results as an indicator of competence, and increase the cost of taking and administering exams for students and teachers.