# The prevalence of breast cancer in women over 40 in country X is estimated to be 0.8% (i.e., 8 in ev

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.
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?
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Ah, the classical cancer detection problem. Let's use a large sample for this one. Say, 1,000,000 women.
Out of these 1,000,000 women:
8,000 will have cancer
7,200 (90% of 8,000) of them will get a positive test result
800 of them will have a negative result (a false negative)
992,000 will NOT have cancer
74,400 (7.5% of 992,000) of them will get a (false) positive test result
917,600 will get a negative result
So, out of the 81,600 (7,200+74,400) women who get a positive result, only 7,200 of them actually have cancer. Meaning that if your test is positive you have a 9.12% probability to have cancer.
Now, if we gather these 81,600 women:
From the 7,200 having cancer :
6,480 will get a second positive result
720 will get a negative one
From the 74,400 NOT having cancer :
5,580 will get a second (false) positive result
68,820 will get a negative one
To conclude, out of the 12,060 women with a double positive result, 6,480 (53,7%) will actually have cancer. Conversely, out of the 69,540 women having a positive, then negative result, 68,820 (99,0%) actually do NOT have cancer.