4-3 Complements, Conditional Probability, and Bayes’ Theorem 173 The preceding formula is a formal expression of conditional probability, but blind use of formulas is not recommended. Instead, we recommend the intuitive approach, as illustrated in Example 2. FORMAL APPROACH FOR FINDING P1 B∣ A2 The probability P1B A2 can be found by dividing the probability of events A and B both occurring by the probability of event A: P1B A2 = P1A andB2 P1A2 Pre-Employment Drug Screening EXAMPLE 2 Refer to Table 4-1 to find the following: a. If 1 of the 555 test subjects is randomly selected, find the probability that the subject had a positive test result, given that the subject actually uses drugs. That is, find P(positive test result subject uses drugs). b. If 1 of the 555 test subjects is randomly selected, find the probability that the subject actually uses drugs, given that he or she had a positive test result. That is, find P(subject uses drugs positive test result). TABLE 4-1 Results from Drug Tests of Job Applicants Positive Test Result (Test shows drug use.) Negative Test Result (Test shows no drug use.) Subject Uses Drugs 45 (True Positive) 5 (False Negative) Subject Does Not Use Drugs 25 (False Positive) 480 (True Negative) SOLUTION a. Intuitive Approach: We want P(positive test result subject uses drugs), the probability of getting someone with a positive test result, given that the selected subject uses drugs. Here is the key point: If we assume that the selected subject actually uses drugs, we are dealing only with the 50 subjects in the first row of Table 4-1. Among those 50 subjects, 45 had positive test results, so we get this result: P1positive test result subject uses drugs2 = 45 50 = 0.900 Formal Approach: The same result can be found by using the formula for P1B A2 given with the formal approach. We use the following notation. P1B A2 = P1positive test result subject uses drugs2 where B = positive test result and A = subject uses drugs. In the following calculation, we use P(subject uses drugs and had a positive test result) = 45>555 and P(subject uses drugs) = 50>555 to get the following results: P1B A2 = P1A and B2 P1A2 Group Testing During World War II, the U.S. Army tested for syphilis by giving each soldier an individual blood test that was analyzed separately. One researcher suggested mixing pairs of blood samples. After the mixed pairs were tested, those with syphilis could be identified by retesting the few blood samples that were in the pairs that tested positive. Since the total number of analyses was reduced by pairing blood specimens, why not combine them in groups of three or four or more? This technique of combining samples in groups and retesting only those groups that test positive is known as group testing or pooled testing, or composite testing. University of Nebraska statistician Christopher Bilder wrote an article about this topic in Chance magazine, and he cited some real applications. He noted that the American Red Cross uses group testing to screen for specific diseases, such as hepatitis, and group testing is used by veterinarians when cattle are tested for the bovine viral diarrhea virus. continued
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