In response to my post on random drug testing in Florida, commenter Scott mentioned the danger of false positives and said that a second test in his case cleared him of having taken drugs. My advice to anyone is that whenever you take a high-stakes test for anything that has a small incidence in the general population (drugs, diseases, whatever) and it comes out positive, always consider asking for a second test.
Suppose the percentage of drug users in the general population is (say) about 1 in 100. You are told that there is a test that is pretty good in that it has a ‘false positive’ rate of only 5%, meaning that if a randomly selected group is tested, only 5% of the people who do not use drugs will have test results that come out positive. Also you are told that the false negative rate is negligible, meaning that if someone does take drugs, the test will almost certainly not come out negative.
Suppose someone takes part in this random testing and the result is positive. What do you think are the chances that the person has taken drugs? Most people would think that it is very high. They may put it as high as 95%, thinking that if there is a 5% false positive rate and 0% false negative rate, that means that the likelihood of someone testing positive actually having taken drugs is 95%. This sounds eminently reasonable but the actual chance is just 1 in 6 or less than 17%!
How come? This becomes easier to understand if we shift from talking in terms of probabilities (which are not intuitive) to talking about numbers. Suppose you are one of 1000 people being randomly tested. (Any size will do. I have chosen 1000 because it is a nice round number.) Then an incidence of drug use of 1 in 100 means that we expect 10 people to actually have taken drugs and thus test positive, and 990 to be drug-free.
But a 5% false positive rate will result in about 50 of the 990 people who do not take drugs also testing positive. So a total of 60 people will test positive, of whom only 10 will actually have taken drugs. So even with a false positive rate of only 5%, the chances are 5 out of 6 or 83% that a person who tests positive is actually drug-free.
What the positive test result has done is increase in the odds of the person having taken drugs from 1 in 100 (or 1%) to 1 in 6 (or slightly less than 17%). But there is still an 83% chance that the person has not taken drugs.
So always consider asking for data about the rates of incidence of drug use (or disease) and the rate of false positives to do your own calculations and consider a repeat test before making any major decision based on such tests.