A rare, but fatal, disease afflicts 1 person in 100,000. Researchers have developed a powerful diagnostic test for this disease. Everyone who has the disease will test positive. Ninety-nine percent of those who do not have the disease will test negative and one percent will test positive. Harold Dilemma was given this test during a routine physical examination and he tested positive. Harold was horrified.
(a) Harold has read about an available surgical procedure. His insurance will cover the financial cost of this surgery. The surgery would certainly eliminate the disease if he has it, but whether or not he has the disease, there is a probability of 1/200 that he would not survive the surgery. Before doing detailed calculations, do you think that undergoing the surgery would increase his overall probability of survival? _______________
(b) To find the probability that Harold actually has the disease, given that he tests positive for it, let us reason as follows. The disease afflicts 1 person in 100,000, so in a population of 1,000,000 people, the number of people who have the disease can be expected to be about
_____________.
Suppose that the test is administered to all 1,000,000 people. Given that one percent of those who do not have the disease will test positive, the total number of people who test positive for the disease can be expected to be about __________ Therefore, of all those who test positive for the disease, the fraction who actually have it is ___________ So, given that he tests positive, what is the probability that Harold has the disease? __________ Would Harold improve his survival probability by undergoing surgery? Explain. __________
(c) Suppose that this disease afflicted one person in 10,000 rather than one in 100,000. Then if Harold tested positive, what would be the probability that he has the disease? _________ Would he then improve his survival probability by undergoing the surgery? _______