When are Conditional Probabilities Equal?

One of the simplest inferential mistakes is to assume a conditional probability is the same as its inverse. For example, around 75% of criminals are male, but it would be a mistake to assume that 75% of males are criminal. This seems fairly obvious. Yet on the level of intuitive reasoning, we don't seem to be particularly effective at keeping the two distinct, as documented in (for example) Villejoubert and Mandel (2002). A fallacy of this sort might have lain behind the UK's controversial counter-terrorism campaign in 2008: even assuming that the majority of terrorists exhibit suspicious behaviour, it doesn't remotely follow that the majority of people exhibiting suspicious behaviour are terrorists.

It's therefore worth remembering when inverse probabilities are equal to one another, so as to take inferential precautions when they are not. A simple implication of Bayes' theorem is that the conditional probability of A given B is similar to the conditional probability of B given A when the probabilities of A and B are close to one another. If there were approximately the same number of terrorists as there were people behaving suspiciously, and most terrorists behaved suspiciously, then one could infer that most suspiciously-behaving people were terrorists.

The opposite is also true. If there is a big discrepancy between the probabilities of A and B, the conditional probabilities will also differ significantly, the exception being when there is zero probability that A and B can be true together.