Or, if the false positives are probabilistically independent, just run the same test again. If your test gives back 1% false positives, running it twice gets you down to .01%, three times gives you one in a million.
Unfortunately, this is essentially never the case. The reason is that if the outcomes were probabilistically independent, then you would just declare a standard protocol of always running the test N times, and call that the test. It's safe to assume that such trivial optimizations have already been done for medical tests.
The common syphilis test looks for antibodies that are produced in response to the bacteria. However, the body may produce those antibodies (on a continuing basis) for other reasons. If you find these antibodies the first time in a given person, you'll keep finding them when you re-test.
