Consider two medical tests, A and B, for a virus. Test A is 95% eective at recog
ID: 3624851 • Letter: C
Question
Consider two medical tests, A and B, for a virus. Test A is 95% eective at recognizingthe virus when it is present, but it has a 10% false positive rate (indicating the virus
is present, when it is not). Test B is 90% eective recognizing the virus, but has a 5%
false positive rate. The two tests use independent methods of identifying the virus.
The virus is carried by 1% of all people. Say that a person is tested for the virus using
only one of the test, and that test comes back positive for carrying the virus. Which
test returning positive is more indicative of someone really carrying the virus? Justify
your answer mathematically
Explanation / Answer
Let V be the statement that the patient has the virus, and A and B the statements that the medical tests A and B returned positive, respectively. The problem statement gives: P(V ) = 0.01 P(A|V ) = 0.95 P(A|¬V ) = 0.10 P(B|V ) = 0.90 P(B|¬V ) = 0.05 The test whose positive result is more indicative of the virus being present is the one whose posterior probability, P(V |A) or P(V |B) is largest. One can compute these probabilities directly from the information given, finding that P(V |A) = 0.0876 and P(V |B) = 0.1538, so B is more indicative. Equivalently, the questions is asking which test has the highest posterior odds ratio P(V |A)/P(¬V |A). From the odd form of Bayes theorem: P(V |A)/P(¬V |A)=P(A|V )/P(A|¬V ) P(V )/P(¬V ) we see that the ordering is independent of the probability of V , and that we just need to compare the likelihood ratios P(A|V )/P(A|¬V ) = 9.5 and P(B|V )/P(V |¬V ) = 18 to find the answer.