The prevalence of breast cancer in women over 40 in country X is estimated to be $0.8\%$ (i.e., $8$ in every $1,000$ women in that age group).
Mammograms test for the presence of breast cancer. A positive result indicates that the disease is present. A negative result indicates that it is not.
The sensitivity of a mammogram test for breast cancer is estimated to be $90\%$. This is the probability that the mammogram will give a positive result when the person being tested does have breast cancer.
The false positive rate for the mammogram is $7.5\%$. This is the probability that the mammogram will give a positive test result when the person being tested does not have breast cancer.
(At this point, there are $816$ women who test positive. I've calculated that myself ;) ) All women who test positive ($816$) in the mammogram are referred for a further, different examination, which however has the same sensitivity and false positive rates as the first test.
What is the probability that a woman referred for this examination and testing positive again, actually does have breast cancer?
What is the probability that a woman referred for this second examination and testing negative this time, actually does not have breast cancer?
Thanks so much. I'm struggling.