Here’s how you solve this kind of question using Bayes Theorem:

1. Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)

2. However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected

3. In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a “false positive” (9% of 98.8 = 8.892)

4. So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)

5. Of these 9.948, only 1.056 will actually have breast cancer

6. 10.6% = (1.056/1.056+9.892)    ç This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem: