If the following paragraph doesn’t cause you to nod knowingly, you really should read the whole article. (BTW, I’ve changed the figures to percentages, for enhanced legibility by non mathematicians).
Suppose that NSA’s system is really, really, really good, really, really good, with an accuracy rate of 90%, and a misidentification rate of .001%, which means that only 3,000 innocent people are misidentified as terrorists. With these suppositions, then the probability that people are terrorists given that NSA’s system of surveillance identifies them as terrorists is only 23%, which is far from 100% and well below flipping a coin. NSA’s domestic monitoring of everyone’s email and phone calls is useless for finding terrorists.
This kind of result is often very suprising and non-intuitive, and hence important. When reality diverges from “common sense”, we need to understand why, so we can explain it to people who like to trust “common sense” in their decision making processes (eg Daily Mail readers ;-) ). This kind of result crops up all over the place… I first came across it in the context of medical diagnosis, where it basically explains why misdiagnosis happens so often. Quite simply, the numbers are just stacked against us. There’s nothing we can do about it – we just have to understand what’s happening and get on with it.