21Bayes' theorem on disease test

A disease affects 1% of people. A test detects it 99% of the time and gives a false positive 5% of the time. Given a positive test, find $\displaystyle P(\text{disease})$. Give the answer to 4 decimal places.

Most people testing positive are actually healthy, since the disease is rare. Build the tree: P(D∩+) = 0.01×0.99 = 0.0099, P(not D∩+) = 0.99×0.05 = 0.0495, so P(+) = 0.0594. Then P(D|+) = 0.0099/0.0594 ≈ 0.1667.