Math for doctors:

I'll give you a couple examples relating to medical care. In the U.S. and many European countries, women who are 40 years old are told to participate in mammography screening. Say that a woman takes her first mammogram and it comes out positive. She might ask the physician, "What does that mean? Do I have breast cancer? Or are my chances of having it 99%, 95%, or 90% ­ or only 50%? What do we know at this point?" I have put the same question to radiologists who have done mammography screening for 20 or 25 years, including chiefs of departments. A third said they would tell this woman that, given a positive mammogram, her chance of having breast cancer is 90%.

However, what happens when they get additional relevant information? The chance that a woman in this age group has cancer is roughly1%. If a woman has breast cancer, the probability that she will test positive on a mammogram is 90%. If a woman does not have breast cancer the probability that she nevertheless tests positive is some 9%. In technical terms you have a base rate of 1%, a sensitivity or hit rate of 90%, and a false positive rate of about 9%. So, how do you answer this woman who's just tested positive for cancer? As I just said, about a third of the physicians thinks it's 90%, another third thinks the answer should be something between 50% and 80%, and another third thinks the answer is between 1% and 10%. Again, these are professionals with many years of experience. It's hard to imagine a larger variability in physicians' judgments — between 1% and 90% — and if patients knew about this variability, they would not be very happy. This situation is typical of what we know from laboratory experiments: namely, that when people encounter probabilities — which are technically conditional probabilities — their minds are clouded when they try to make an inference.

What we do is to teach these physicians tools that change the representation so that they can see through the problem. We don't send them to a statistics course, since they wouldn't have the time to go in the first place, and most likely they wouldn't understand it because they would be taught probabilities again. But how can we help them to understand the situation?

Let's change the representation using natural frequencies, as if the physician would have observed these patients him- or herself. One can communicate the same information in the following, much more simple way. Think about 100 women. One of them has breast cancer. This was the 1%. She likely tests positive; that's the 90%. Out of 99 who do not have breast cancer another 9 or 10 will test positive. So we have one in 9 or 10 who tests positive. How many of them actually has cancer? One out of ten. That's not 90%, that's not 50%, that's one out of ten.

Here we have a method that enables physicians to see through the fog just by changing the representation, turning their innumeracy into insight. Many of these physicians have carried this innumeracy around for decades and have tried to hide it. When we interview them, they obviously admit it, saying, "I don't know what to do with these numbers. I always confuse these things." Here we have a chance to use very simple tools to help those patients and physicians to understand what the risks are and which enable them to have a reasonable reaction to what to do. If you take the perspective of a patient — that this test means that there is a 90% chance you have cancer — you can imagine what emotions set in, emotions that do not help her to reason the right way. But informing her that only one out of ten women who tests positive actually has cancer would help her to have a cooler attitude and to make more reasonable decisions.