While it is not surprising, it is worrisome that doctors have trouble with statistics, particularly conditional probabilities. 25 German doctors were asked about the following situation. It is clearly a tricky question, but it is surely a type of question that doctors are exposed to constantly:
The probability that one of these women has breast cancer is 0.8 percent. If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram. If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram. Imagine a woman who has a positive mammogram. What is the probability that she actually has breast cancer?
The results of this small trial were not encouraging:
[The] estimates whipsawed from 1 percent to 90 percent. Eight of them thought the chances were 10 percent or less, 8 more said 90 percent, and the remaining 8 guessed somewhere between 50 and 80 percent. Imagine how upsetting it would be as a patient to hear such divergent opinions.
As for the American doctors, 95 out of 100 estimated the womanâ€™s probability of having breast cancer to be somewhere around 75 percent.
The right answer is 9 percent.
You would think that this sort of quantitative analysis would play an important role in the medical profession. I am certain that a great many people around the world have received inappropriate treatment or taken unnecessary risks because doctors have failed to properly apply Bayes’ Theorem. Indeed, false positives in medical tests are a very commonly used example of where medical statistics can be confusing. It is also a problem for biometric security protocols, useful for filtering spam email, and a common source of general statistical errors.
The proper remedy for this is probably to provide doctors with simple-to-use tools that allow them to go from data of the kind in the original question to a correct analysis of probabilities. The first linked article also provides a good example of a more intuitive way to think about conditional probabilities.