A new article at Behavioral and Brain Sciences reviews attempts to explain the following puzzle. People do badly at questions worded this way:
The probability of breast cancer is 1% for a woman at age forty who participates in routine screening. If a woman has breast cancer, the probability is 80% that she will get a positive mammography. If a woman does not have breast cancer, the probability is 9.6% that she will also get a positive mammography. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer? __%
They do much better at questions worded this way:
10 out of every 1,000 women at age forty who participate in routine screening have breast cancer. 8 out of every 10 women with breast cancer will get a positive mammography. 95 out of every 990 women without breast cancer will also get a positive mammography. Here is a new representative sample of women at age forty who got a positive mammography in routine screening. How many of these women do you expect to actually have breast cancer? ___ out of ___.
Whatever the explanation, the lesson should be clear: prefer to reason in terms of frequencies, instead of probabilities. Thanks to Keith Henson for the pointer.