Question: A disease is present in 5 out of 100 people, and a test that is 90% accurate is administered to 100 people. If one person in the group tests positive, what is the probability that this one person has the disease?

My answer:

There is not enough information to be able to answer because it is not clear what “90% accurate” means. You could have high specificity and low sensitivity or vice versa, and the answer would be different. In addition, if only one person out of 100 in this sample tests positive, it is a strong indication that this sample is not the same as the population that has 5% positive.

We can get rid of the second problem by testing just one person and assuming the person is from the same population. The first problem is not so easy to solve.

Situation 1

True positives – 5

True negatives – 85

False positives – 10

False negatives – 0

Situation 2

True positives – 0

True negatives – 90

False positives – 5

False negatives – 5

In situation 1, 15% of people test positive. Of these, 5 are actually positive. So, the answer would be 0.33 or 33%.

In situation 2, 5% of people test positive. Of these, 0 are actually positive. So the answer would be 0.