Probability and Medicine
When he explained that she would either respond or not respond, she became confused and said "You mean my chances of getting better are really only 50%?" Clearly, she was mistaking the binary aspect of the treatment outcome (getting better or not) with the probability that she herself would get a response.
Friedman then speculated on why her patient might have had this misconception. He points out that mathematicians have attributed such problems on innumeracy "the arithmetic equivalent of illiteracy". He also mentioned that some misunderstanding might arise from a natural human tendency to not attribute bad (or any striking) events to chance.
Personally, I think the example Friedman cited has more to do with innumeracy. However, I don't like the word because of its pejorative connotation. The fact is that most of us have this type of innumeracy even doctors (if you can believe it). Probability is one of those terribly difficult philosophical problems that trouble just about all of us.
The dynamics of a clinical situation will determine the probability of a given patient developing a specific disease. A smoker has a higher probability of getting lung cancer than a nonsmoker, but an individual will either get it or not period. This sounds straight forward but a lot of people have problems with it. Some smokers never get lung cancer and some nonsmokers do. The reason is that smoking is not the only factor that leads to lung cancer. The more factors we understand (for example age, exposure to other toxins such as oxidants and genetics), the more precise the probability estimate will become.
This becomes very important as physicians increasingly embrace evidence-based medicine (EBM). In the desire to cite statistics of medical outcomes (such as the chance of developing a certain disease or the likelihood that a certain treatment will work) it is very important to recognize that every patient is different. The study population of a particular study will surely have a cross-section of many different types of participants. The patient's observed probability will be closer to patients more like himself -- maybe closer in ways that weren't even imagined or assessed by the researchers.
The original studies looking at the impact of cholesterol on cardiac outcomes didn't subdivide patients by measuring the different types of cholesterol such as LDL, HDL or triglycerides. Had they done so, individual probabilities of adverse outcomes could have been better stratified (as they have been subsequently).
As physicians, we have to do a better job of explaining these concepts to our patients. At the same time, we need to do a better job of understanding them ourselves! What's true in a study may not be true for a particular patient.
I want to close this post with my favorite probability brainteaser: If you flip a coin nine times and it comes up heads each time, what is the probability that it will come up heads the tenth time? I'll put the correct answer as the first comment to this post.