1. The problem statement, all variables and given/known data A diagnostic test is used to detect the HIV virus. It is known that 2% of the people in a city have the virus. Extensive research on the diagnostic test reveals that its results are correct 95% of the time. In other words, whether or not an individual has the HIV virus, the probability that the test gives a correct diagnosis is 0.95. (a) A person took the test and was diagnosed to have the HIV virus. What is the probability that he actually did not have the virus? (b) Three persons took the test and were all diagnosed to have the HIV virus. What is the probability that at least two of them have the virus. (Answers: (a) 0.7206 (b) 0.1906) 2. Relevant equations Probability Formulae 3. The attempt at a solution (a) 0.98 x 0.05 / (0.98 x 0.05 + 0.02 x 0.95) = 0.720588 (b) 1 - (0.7206)3 - 3 x (0.7206)2 x (1 - 0.7206) = 0.19057 For part (a), I don't know why the answer can be calculated from the above formula. Can anyone explain it to me? Tree Diagram: B1 - HIV B1.1 - Y B1.2 - N B2 - No HIV B2.1 - Y B2.2 - N B : Branch; Y : Correct Diagnosis, N : Incorrect Diagnosis In addition, may the above formula conflict with the tree diagram I drew? Thank you very much!