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: 
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)
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.