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In this section we introduce an important result that has many applications to manage- ment decision making. Bayes' theorem provides a way of revising conditional probabil- ities by using available information. It also provides a procedure for determining how probability statements should be adjusted, given additional information. Reverend Thomas Bayes (1702-1761) developed Bayes' theorem, originally published in 1763 after his death and again in 1958 (Bayes 1958). Because games of chanceand, hence, probability-were considered to be works of the devil, the results were not widely publicized. Since World War II a major area of statistics and a major area of management decision theory have developed based on the original works of Thomas Bayes. We begin our development with an example problem, followed by a more formal development, Example 3.23 Drug Screening (Bayes' Theorem) A number of amateur and professional sports organizations use routine screening tests to determine if athletes are using performance-enhancing drugs. Jennifer Smith, presi- dent of an amateur athletic union, has asked you to determine the feasibility of screen ing athletes to determine if they are using performance-enhancing drugs. Amateur athletes are increasingly denied participation or deprived of victories if they are found to be users. As part of the study, you propose the following scenario for analysis. Suppose that 10% of the athletes seeking participation in the athletic union have used performance- enhancing drugs. In addition, suppose that a test is available that correctly identifies an athlete's drug usage 90% of the time. If an athlete is a drug user, the probability is 0.90 that the athlete is correctly identified by the test as a drug user. Similarly, if the athlete obability t3 is not a drug user, the probability is 0.90 that the athlete is correctly identified as not using performance-enhancing drugs. We should note that there are potential ethical and possible legal questions con- cerning the use of these tests. Here, we are concerned about the feasibility of using such a test if one has decided that such a test is proper, given the legal and value systems