Updating direct draw
Here's an alternate version of the problem on which doctors fare somewhat better: 10 out of 1000 women at age forty who participate in routine screening have breast cancer.
The most common mistake is to ignore the original fraction of women with breast cancer, and the fraction of women without breast cancer who receive false positives, and focus only on the fraction of women with breast cancer who get positive results.
Expressed as a proportion, this is 80/1,030 or 0.07767 or 7.8%.
To put it another way, before the mammography screening, the 10,000 women can be divided into two groups: As you can check, the sum of all four groups is still 10,000.
The proportion of cancer patients with positive results, within the group of patients with positive results, is the proportion of (A) within (A C): 80 / (80 950) = 80 / 1030 = 7.8%.
If you administer a mammography to 10,000 patients, then out of the 1030 with positive mammographies, 80 of those positive-mammography patients will have cancer.
100 out of 10,000 women at age forty who participate in routine screening have breast cancer.