Problem 2 An obstetrician orders a prenatal test for a rare genetic disorder. We let A be the event that the fetus has that abnormality, which is known to occur with probability P(A)=1/20,000=0.005%. A traditional screening test gives a (correct) positive result for that disorder with 98% probability if it is in fact present, and a (false) positive result with 2% probability when it is not. It is also possible to conduct a costlier but more precice test which reconstructs fetal DNA from the mother's blood sample. That second test has a false positive rate equal to zero, and detects the abnormality with probability 99.9%. We denote the event of a (correct or false) detection in the first test with B1, and that of a detection in the second test with B2, and we assume that B1 and B2 are independent conditional on A. (a) Restate the probabilities of false and correct positives as conditional probabilities. (b) Use the law of total probability to compute the probability of a positive result for the traditional screening test, P(B1). (c) Derive the conditional probability of the fetus having the genetic disorder (event A ) given a positive result in the first test (event B1 ). (d) Suppose that after receiving a positive result on the first, traditional test, the ob-gyn orders the DNA analysis which then yields a negative result. Derive the conditional probability of A given B1B2C. (e) Now suppose that the patient first receives a negative result from the DNA analysis, but then a positive result for the screening test. Derive P(AB2C) and use Bayes theorem to compute the conditional probability of A given B1B2C. How does your result compare to your answer in (d)? (f) Compare your results for P(AB1) and P(AB1B2C). What does this mean for interpreting a positive result for the screening test after having tested negative using the second, more precise test? Discuss