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.
Perhaps this line of thinking is correct?:

For any random person, there is a 1% chance that they have cancer, and so there is then a $.01 \* 1$ chance that the mammogram of a random person will be positive. If they do not have cancer, there is a 1% chance that the mammogram will be positive. 

So it is intuitively close to a coin flip for a random person. I'm not sure how to explain the additional $0.0025$ in favor of cancer given a positive mammogram.