"What is the probability that a woman with a positive mammography result actually has breast cancer?"

Been reading Statistics Done Wrong and it pointed me to a paper that addressed the question that's the title of this post:

Conditional probabilities: The probability that a woman has breast cancer is 0.8%. If she has breast cancer the probability that a mammogram will show a positive result is 90%. If a woman does not have breast cancer the probability of a positive result is 7%. Take, for example, a woman who has a positive result. What is the probability that she actually has breast cancer?

... We see quickly that only seven of the 77 women who test positive actually have breast cancer, which is one in 11 (9%).

How good do the authors find doctors to be at interpreting such statistics?

When asked to estimate the probability that a woman with a positive screening mammogram actually has breast cancer, doctors who received conditional probabilities gave answers that ranged from 1% to 90%; very few of them gave the correct answer of about 8%.

The authors of Statistics Done Wrong have a little more to say on how well people interprete such data:

If you administer questions like this one to statistics students and scientific methodology instructors, more than a third fail.8 If you ask doctors, two thirds fail.

I seem to vaguely recall seeing such an example back once-upon-a-time when I took statistics. The answer of 9% might also tell you why medical officials are questioning the value of routine mammograms.

(Hmm... interesting. Realized that the paper separately notes figures of approximately 8% and 9%. Seems that if you look at the data in terms of natural frequencies you wind up with rounding giving you a slightly lower result here).