This question was posed to a group of medical professionals.
A woman is tested for a particular type of cancer, and the test is positive. Worried, she asks how likely it is that she really has cancer. Which of these is the best answer?
A) 9 in 10
B) 8 in 10
C) 1 in 10
D) 1 in 100
You’ll need some information about the test.
1) One in a thousand women actually have this particular cancer.
2) If a woman has the cancer, the test is 90% likely to return positive.
3) If she does not, however, the test still has a 9% chance of returning positive.
This is a typical question given to students of probability theory, when they learn abot Bayes’ theorem. As you can see, it’s also an important question for medical professionals and their patients. Alas, only 21% of the professionals surveyed got the answer correct. They’d have done better if they just randomly guessed.
It’s no doubt not practical to teach medical professionals Bayes’ theorem. After all, they have enough to learn already, to keep track of rapid progress in their field. This fascinating article from the BBC outlines this example (and gives a nice chart to help you understand why the correct answer is C), and other examples.
Such as the example of breast cancer screening by mammography.
According to the statistics the aricle quotes, 5 in 1000 unscreened women die from breast cancer. Screening reduces this to 4 in 1000. Is that a 20% reduction in mortality? Or a 0.1%? Or both, depending on what you mean?
It’s important to know what statistics mean and how probability works, or at least to know that you don’t know how they work.