Bayes' Rule can be applied to situations involving the responses to sensitive questions where a respondent may not answer truthfully due to embarrassment or incrimination. For example, a university administrator may ask students whether they have ever cheated on a final exam for a college course. Instead of answering the question directly, each student rolls a die. If a 1 or 2 appears, then the student answers the question truthfully (a “yes” or “no” answer). If a 3, 4, 5, or 6 appears, then the student lies. Hence, if a student lies, then the student answers “yes” if s/ he has never cheated, and answers “no” if s/ he has ever cheated on a final exam. Since the administrator does not see the outcome of the roll of the die, he does not know if the student is answering the “cheating” question truthfully. Suppose the administrator uses this procedure to solicit responses from 500 randomly chosen students. Of those individuals, 225 of them answer “yes.”
a. If Chris is among those randomly selected students, what is the probability that he has ever cheated on a final exam? Use the law of total probability to find an equation for P(yes) in terms of P(cheated on final) and then solve for P(cheated on final).
b. If Chris is one of those people who answered “yes,” what is the probability that he has ever cheated on a final exam?