A test of a disease presents a rate of 5% false positives. The disease strikes 1/1,000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient’s test is positive. What is the probability of the patient being stricken with the disease?
Most doctors answered 95%, simply taking into account the fact that the test has a 95% accuracy rate.
The answer is the conditional probability that the patient is sick and the test shows it - close to 2%. Less than one in five professionals got it right.
As for the answer. Assume there are no false negatives. Consider that out of 1,000 patients who are administered the test, one will be expected to be afflicted with the disease. Out of a population of the remaining 999 healthy patients, the test will identify about 50 to have the disease. (it is 95% accurate). The correct answer should be that the probability of being afflicted with the disease or someone selected at random from those with a positive test is:
= number of afflicted persons / number of true and false positives = 1/51
Alarming, isn’t it? If my doctor cannot interpret the test correctly, how can I trust him or her to treat me properly? I might actually do better to try to heal myself.
I found this case in “Fooled by Randomness” by Nassim Nicholas Taleb. An excellent book full of examples of the role of chance in life. It also tries to explain why we misinterpret probability. Fascinating and highly recommended.