Ever been asked to settle something beyond a reasonable doubt? Ever taken part in a cause in which the evidence is overwhelming? Ever been completely sure you're right? Sure you have. Plenty of times. So many times that occasionally, you have to have been wrong.
What would you consider certainty? In an ideal world, 100 percent certainty would be the only acceptable definition, but as we don't live in an ideal world, settling for nothing less than perfect certainty means not a lot would get done. And so we relax our standards. We don't have to relax them much, but unfortunately, even a little bit makes a difference eventually.
Let's say we set up a test that gives establishes what we want to find out with a certainty of 99 percent. This test could establish the guilt or innocence of a suspect, or the presence of a medical condition. It could also establish more complicated things, like the likelihood that a certain number of crimes in an area are committed by one person, or the likelihood that a particular drug is effective. All these things can be established with a 99 percent certainty. Or, rather, any one of these things can be established with a 99 percent certainty. What if we want to establish all four of these things?
The math isn't complicated. The likelihood of a coin flip coming up heads is 0.5. The likelihood of a coin flip coming up heads two times in a row is 0.5 multiplied by 0.5, or 0.25. Since a 99 percent probability is 0.99, we can multiply it by itself the appropriate number of times to get roughly 0.96, or 96 percent. We are 96 percent certain that all four of the tests are right. That sounds pretty good, except all of those tests are done hundreds of times every year. Repeated 100 times, the probability of the tests being right falls to 37 percent. Repeated 300 times, and the probability falls to 5 percent. By repetition 450, and we're down to 1 percent certainty. In other words, we are just as certain that we have got something wrong as we were certain, in each individual case, that we got it right.
This idea gets a lot of flack, because of the unfortunately large number of people who decide that, "sometimes tests are wrong," means, "I get to decide that any result I don't like is wrong." The point is to let us know the real odds that sometimes the seemingly incontrovertible test results we get are wrong, and to understand why there is a failure in these tests. No one has to screw up. No test has to be shoddy. No single case has to be weak. Do something enough, and someone will beat the odds.
[Via The Improbability Principle.]