Question

A certain test for a particular type of cancer is known to be 95% accurate. A person takes the test and the results are positive. Suppose the person comes from a population of 100,000 where 2000 people suffer from that disease.

What if the person takes a second test and result is still positive, what can we conclude about his odds of having cancer?

Answer 1

It is given that a certain test for a particular type of cancer is known to be 95% accurate. Thus, let the events {T>0} stand for the test being positive and {T<0} stand for the test being negative.

Let the set of healthy patients be H and cancer patients be C.

Thus, P(T>0|C) = 0.95 and P(T>0|H) = 0.05

P(T<0|C) = 0.05 and P(T<0|H) = 0.95

Thus, we have 2000 cancer patients and 98000 healthy people as the total population is 100,000.

Thus, the probability that a person chosen at random is healthy = 98000/100000 = 0.98 and the probability that the person suffers from cancer = 2000/100000 = 0.02

Thus, using Bayes' theorem, the probability that the person suffers from cancer given the test is positive is,

P(C|T>0) = [P(T>0|C)P(C)]/[P(T>0)]

= [P(T>0|C)P(C)]/[P(T>0|C)P(C)+P(T>0|H)P(H)]

= (0.95*0.02)/[(0.95*0.02)+(0.05*0.98)] = 0.278

This result states that if the person takes a second test and result is still positive, then we can conclude that there is 27.8% chance of having cancer.