The human mind isn't attuned to statistical argument. Suppose a person selected at random on a US street tests positive for HIV: what's the chance that they really are infected with the virus? Know that the test is very accurate, yielding only a 0.0001 fraction of false positives. Given this number, it comes as a surprise to most people — physicists too, I suspect — that this random individual, after testing positive, actually has only a 50% chance of really being infected.
We're lured into error by that impressively small number, which leads us, almost without thinking, to conclude 'very unlikely'. A little care with the conscious mind says that we need to include the fact that our 'random' individual, before testing, has only a 1 in 10,000 chance of being HIV-positive, this being the general US infection rate. Of 10,000 people tested, on average, one will be truly infected, as their test will almost certainly show. But of the other 9,999 without the virus, on average one will also turn up a false positive. Hence, half the positives will be false (the results change markedly if the person isn't selected 'at random', but belongs to a group already known to be at high risk for HIV infection.)
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