Social psychologist Gird Gigerenzer's new book covers some instances of asking doctors to give probabilities to their patients, and they do a horrible job. The question presents the information exactly as doctors are taught: prevalence, sensitivity, and specificity (false positives).
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?
A prestigious doctor, department chief with 30 years of experience "was visibly nervous while trying to figure out what he would tell the woman. After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent. Nervously, he added, ‘Oh, what nonsense. I can’t do this. You should test my daughter; she is studying medicine.’ He knew that his estimate was wrong, but he did not know how to reason better. Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities."
And he was typical: more than 90% of the doctors were wrong, mostly very wrong.
When the question was posed in terms that are easier for people to understand, nearly all of the doctors got the question right.