Question: I'm trying to wrap my head around the result of Bayes Theorem applied to the classic mammogram example, with the twist of the mammogram being perfect.

That is,

Incidence of cancer: $.01$

Probability of a positive mammogram, given the patient has cancer: $1$

Probability of a positive mammogram, given the patient does not have cancer: $.01$

By Bayes:

P(cancer | mammogram+) = $\dfrac {1 \cdot .01}{(1 \cdot .01) + (.091 \cdot .99)}$

$ = .5025$

So, if a random person from the population takes the mammogram, and obtains a positive result, there is a 50% chance they have cancer? I'm failing to intuitively understand how the tiny 1% chance of a false positive in 1% of the population can trigger a 50% result. Logically, I would think a perfectly true positive mammogram with a tiny false positive rate would be much more accurate.

Answer: 
A key problem with mammography that hasn't been adequately addressed in the discourse is the faulty definition of "positive". This is described in the *Diagnosis* chapter in <http://biostat.mc.vanderbilt.edu/ClinStat> - see the link for *Biostatistics in Biomedical Research* there.

One of the most widely used diagnostic coding systems in mammography is the BI-RADS score, and a score of 4 is a frequent "positive" result. The definition of category 4 is "Not characteristic of breast cancer, but reasonable probability of being malignant (3 to 94%); biopsy should be considered." With a risk range that goes all the way from 0.03 to 0.94 for **one category**, i.e., incredible heterogeneity in what "positive" really means, it is no wonder that we have a mess on our hands.

It is also a sign of unclear thinking that the BI-RADS system has no category for someone with an estimated risk of 0.945.

As Nate Silver so eloquently argues in *The Signal and the Noise*, if we were to think probabilistically we would make better decisions all around. Removing terms such as "positive" and "negative" for medical tests would remove false positives and false negatives and convey uncertainty (and justification for more tests before making a diagnosis) optimally.
There is a nice discussion of this in the book [Calculated Risks](https://booko.com.au/works/41003)

Much of the book is about finding clearer ways of talking about, and thinking about, probability and risk. An example:

The probability that a woman of age 40 has breast cancer is about 1 percent. If she has breast cancer, the probability that she will test positive on a screening mammogram is about 90 percent. If she does not have breast cancer, the probability that she will nevertheless test positive is 9 percent. What are the chances that a woman who tests positive actually has breast cancer?

This is the way the book presents the solution, using 'natural frequencies'. Consider 10,000 women, 1% have cancer so that is 100 women. Of these, 90% will return positive tests (i.e. 90 women with cancer will test positive). Of the 9900 without cancer 9% will return positive test or 891 women. So there are 891 + 90 = 981 women with positive tests of which 90 have cancer. So the chance that a woman with a positive test has cancer is 90/981 = 0.092

If 100% of woman with cancer test positive that just changes the numbers a bit to 100/(100 + 891) = 0.1

[![Natural frequency approach to understanding false positives](https://i.stack.imgur.com/jF6jD.gif)](https://i.stack.imgur.com/jF6jD.gif)