Totally not a statistician, but I'll give it a shot.
For the sake of the argument:
test accuracy: exactly 99.0% accurate
disease incidence: exactly 1 in 1 million
Calculation:
For the sake of simple calculations, let's assume we test exactly 1 million people.
tests positive = (1 * 0.99) + (999999 * 0.01)
tests positive = (.99) + (9999.99)
tests positive = 10000.98
We'll round up for the sake of argument to 10,001 positive results. And we know that only 1 person (remember that we're testing 1 million people) is actually sick. We have 1 actual sick in 10 thousand positives tests. So the probability that the positive test that is right in front of you is actually a truly sick person are 1 in 10 thousand.
Beware, you are making a strong assumption: that the test's accuracy is the same regarding false negatives and false positives. For example, a test may not find enough "anomaly" in an ill person to trigger the positive, thus yielding a false negative; at the same time, it may as well never find any "anomaly" in a sane person, and as a consequence never give any false positive.
Back to your example, it's obvious that a test with a 0.01 probability to give a false positive is completely useless for an illness that affects 1e-6 of people.
Actual descriptions of medical tests routinely give both rates. They often call them "sensitivity" and "specificity". Good luck remembering which is which.
But if only one rate is given, that indicates they're equal. If they're not, then it's reasonable to describe the documentation as incorrect.
Consider a population of 100M people, of which 100 would have the illness. Of them, 99% = 99 would test positive and 1% = 1 would test negative. For the other 99,999,900 healthy people, 99% = 98,999,901 would test negative and 1% = 999,999 would test positive.
In total, 99 + 999,999 people would test positive. Given that a person tests positive, then, there is only a 99 / (99 + 999,999) ~= 0.01% chance that person has the illness.
For sure, this assumes that false positive rate = false negative rate = 1%, but it suffices to illustrate how a highly accurate test can produce misleading results.
A solution would be repeated retesting, as the 1st, 2nd, 3rd, and 4th consecutive positive test results would lead to 0.01%, 1%, 50%, and 99% chances. (Each additional positive test reduces the false positives by 100-fold, whereas the ill patients are very likely to get continually positive results.)
