五、Most medical tests occasionally produce incorrect results, called false
positives and false negatives. When a test is designed to determine whether
a patient has a certain disease, a false positive result indicates that a patient
has the disease when the patient does not have it. A false negative result
indicates that a patient does not have the disease when the patient does have
it.
When large-scale health screenings are performed for diseases with relatively
low incidence, those who develop the screening procedures have to balance
several considerations: the per-person cost of the screening, follow-up costs
for further testing of false positives, and the possibility that people who have
the disease will develop unwarranted confidence in the state of their health.
Consider a medical test that screens for a disease found in 5 people in 1,000.
Suppose that the false positive rate is 3% and the false negative rate is 1%.
Then 99% of the time a person who has the condition tests positive for it, and
97% of the time a person who does not have the condition tests negative for
it.
(a) What is the probability that a randomly chosen person who tests positive
for the disease actually has the disease? (10%)