2. 7 Problem-set problem Let's test our understanding. "A paradigm case of Bayesian reasoning is medical diagnosis? ' [Pinker. Steven. Rationality (p. 150). Penguin Publishing lGroup. Kindle Edition] Economists and psyehologists like to point out how often doctors get this type of reasoning off. {There's like a cottage industry writing papers of this sort.) Suppose that the prevalence. of breast cancer in the population of women is 1 percent. [this means Pr (Hypothesis) 2 0.01]. Suppose that the sensitivity of a breast cancer test (its truepositive rate} is 90 percent [this means Pr {Data|Hypothesis) = 0.9]. Suppose that its falsepositive rate is 9 percent. A woman tests positive. What is the chance. that she has the disease? I will note that Pinker reports that the most popular answer from doctors fails in the range between 80 and 90 percent. 2.8 Problem-set problem This opportunity comes from our recommended textbook. Gamma Chemicals advertises its anabolic steroid detection test as being 95% effective at detecting steroid use. meaning that the test will show a positive result on 95% of all anabolic steroid users. This is P1'{T|A:l = 0.95. where T stands for test positive and A represents use anabolic steroirds. The company also states that its test has a false positive rate of 6%. This means that the probability of a nonuser testing positive is .06. Estimating that. about. 10% of its athletes are using anabolic steroids. which represents Pr (.4) = 0.1. Enormous State University begins testing its football players. The quarterback. Hugo V. Huge. tests positive and is promptly dropped from the team. Hugo claims that he is not using anabolic steroids. How condent can we be that he is not telling the truth? We are interested in the posterior. Pr (MT): What is the probability Hugo used steriods given that the}r tested positive