Question

test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out of 100 people who have the ailment. The test is also 95% accurate for a negative result, meaning that it will correctly exclude the disease in 95 out of 100 people who do not have the ailment. For a certain segment of the population, the incidence of the disease is 4%.

(1) If a person tests positive, find the probability that the person actually has the disease (define appropriate events and use Bayes Theorem);

(2) Now suppose the incidence of the disease is 49%. Compute the probability that the person actually has the disease, given that the test is positive;

(3) The probability you obtained in (1) is much smaller than 0.95, if your computation is correct. Hence, You can conclude that there is only a much smaller probability to claim “the person really has the disease” after knowing that “the test is positive”, though the test has 95% “correctness” Explain this difference

** Population: the segment of the population Experiment: Determine whether or not a selected person has the disease Outcomes: D, Dc(where D and Dc are defined below) Sample space: S = {D, Dc Define necessary events: D = the selected person has the disease; Dc = the selected person does not have the disease. Rate of disease = 4% P(D) = 4% , P(Dc) = 96% E = Testing result shows positive for the selected person Ec = Testing result shows negative for the selected person Correctness of testing: if the selected person does have the disease, the testing will show positive with probability 95%, i.e., (about) 95 positive out of 100 persons who do have the disease. if the selected person does NOT have the disease, the testing will show negative with probability 95%, i.e., (about) 95 positive out of 100 persons who do NOT have the disease. In other words, P(E | D) = 95% and P(Ec| Dc) = 95% Want P(D | E) = ?

Answer 1