I think what you are trying to say is if a test is 90% accurate and we are told that 10% of 1000 people have tested positive for cancer (100 people), then statistically 10 of them would be false-positives, meaning 90 of them truly have cancer

.

Out of the remaining 900 people, 90 of them are false negatives. If you add the two false-populations together (10+90) you get 100 which is 10% of the number we started with. So there is a 10% uncertainty. But of course, we knew that.

Now, we know that there are 90 people that truly have cancer, PLUS we know that out of the people who tested negative, 90 of them are actually positive (have cancer). That means statistically 180 people out of the 1000 tested have cancer (18%).

The test itself only caught 100 (10%) of the cancer patients, so it was only 55.5% (180/100) EFFECTIVE in identifying all those who have cancer within the whole test population. The test itself is still considered to be accurate 90% of the time but saying so can be misleading.

[EDIT]

I’m sure it goes without saying but just wanted to add… the problem is that for either the positive or negative group we don’t know which individuals were falsely identified since it is based on probability.