A doctor gives a patient a test for a particular cancer Befo

A doctor gives a patient a test for a particular cancer. Before the results of the test, the only

evidence the doctor has to go on (i.e.the prior probability of cancer) is that 1 woman in

1,000 has this cancer. Experience has shown that, in 99 percent of the cases in which cancer

is present, the test is positive; and in 95 percent of the cases in which it is not present, it is

negative.

(a)

If the test turns out to be positive, what probability should the doctor assign to the

event that cancer is present? Note: when a group of second-year medical students was

asked this question, over half of the students incorrectly guessed the probability to be

greater than 0.5.

(b)

How large must the prior probability of cancer be to give a posterior probability of 0.5

for cancer given a positive test?

Solution

1 woman in 1000 has cancer.

a) IF test is positive, prob for having cancer = 99/104 = 0.0865

b) The prior prob of cancer should be 0.75