When asked about the accuracy of a mammogram, doctors cite the "false positive rate". Ignore the false positive rate, what patients really need to know is the "positive predictive value" (PPV), that is, the chance of having breast cancer given that one has a positive mammogram.
The PPV for mammography is 9 percent. Nine percent! You heard right. For every 100 patients who test positive, only 9 have breast cancer, the other 91 do not. This may sound like a travesty but it's easily explained: breast cancer is a rare disease, afflicting only 0.8% of women so almost all women who take mammograms do not have cancer, and even a highly accurate test when applied to large numbers of cancer-free women will generate a lot of false alarms.
Junk Charts on Steven Strogatz's piece for the NYT on Bayesian reasoning