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.

Self-plug: Anyone who has trouble teaching Bayes's Theorem to high school students can send them to An Intuitive Explanation of Bayesian Reasoning, which I designed after reading all the pessimistic papers about how hard it is to get subjects to retain Bayes's Theorem for two weeks. Includes neat Java applets.

In teaching AP statistics in high school, I have found that many students have an easier time doing a conditional probability or Bayes' Theorem problem if they put everything in terms of a total frequency of 100.

I think that this has something to do with preferring concrete to abstract, but I'm not sure.