How is it not true? Prior probability has nothing to do with the false positive rate. False positive rate is a function of the inaccuracy of the test.
That pretty clearly depends on the prior probability, P(M).
It doesn't matter what the prior probability is of a random person having used meth (or whatever you are testing for); it only matters how likely it is that someone who has not used meth will be given a positive result.
If that test is performed only on women, the false positive rate will be 100% regardless of accuracy. All positive results would be false positives.
If the test is preformed on any man leaving or entering the oval office, then the false positive rate will likely be significantly less then 100%, as there will be a president tested multiple times in that group, and when we test positive on him, it's not a false positive.
If we test the population at large, most positive results are going to be false positives, simply because we're unlikely to actually have a former president get tested. But in theory the false positive rate could be slightly less then 100% if we got lucky.
As the probability of an actual occurrence in the test group drops to zero, the rate of false positives approaches 100%, regardless of the accuracy of the testing method.
I think you might be talking about the rate relative to all positive trials. I think I am talking about the rate relative to all trials that don't meet the conditions of the test.
So, you survey 100 women and your test tells me that 10 of them have been president, I say that your test has an observed false positive rate of 10%.
If we test a group comprised of only former presidents, all positives must be true positives. So our false positive rate will be 0% while our female test set has been observed to have a false positive rate of 10%. This illustrates that the sample set effects the observed false positive rate.
As the rate of occurrence of what we are testing for approaches 100% in the sample set, the false positive rate drops to zero. Inversely, as the incidence of what we are testing for in a (sufficiently large) sample set decreases from 100%, the false positive rate will eventually increase to some non-zero percentage unless the test is perfect.
So if we only test meth addicts for meth, our false positive rate will be lower then if we test the population at large.