Question

A medical test is not completely accurate. When people who have a certain disease are tested, 90% of them have a "positive" reaction. But 5% of people without the disease also have a "positive" reaction. In a certain city, 20% of the population have the disease. A person from this city is chosen at random and tested; if the reaction is "positive," what is the probability the person has the disease

Answer 1

P(positive) = P(positive | have the disease) * P(have the disease) + P(positive | doesn't have the disease) * P(doesn't have the disease) = 0.9 * 0.2 + 0.05 * 0.8 = 0.22

P(has the disease | positive) = P(positive | has the disease) * P(has the disease)/P(positive) = (0.9 * 0.2)/0.22 = 0.8182