The applications of probability away from dice, casino gambling, and various aspects of natural science can be overlooked. In this chapter, I have picked out some of its appearances in law, social science, sport, and economics to emphasize its ubiquity. The common theme is that decisions we make will depend on the probabilities of various outcomes, so we need methods that lead to reasonably reliable estimates of those different probabilities.
Legal matters
Although Lord Denning, one of the best-known UK judges in the 20th century, had a mathematics degree, few lawyers feel comfortable with probability. This ought to be astonishing, as phrases relating to the subject are used freely in courts. In civil cases, such as libel, to say ‘on the balance of probabilities’ clearly puts the dividing line at 50%. But in criminal cases, where a jury is asked to convict only if they are ‘sure’ of Guilt, there is no consensus on a figure. Some people would wish to convict if they were 80% certain of Guilt, others would use 95% or even higher. These are plainly subjective probabilities. And although the same phrase is used whatever the offence, some would apply a lower threshold of proof for a relatively minor offence. This could make it harder to convict mass murderers than fare dodgers.
Suppose an expert witness testifies that the DNA of the accused matches DNA found at a crime scene, and that the chance of a match between the latter and an innocent person chosen at random is one in several million. Jurors may have two distinct problems with this statement. The first is that they may think that it is equivalent to saying that the chance the crime scene DNA is NOT that of the defendant is one in several million. The second is that they may treat all such tiny figures as equivalent, even though one in ten million differs from one in a billion by a factor of a hundred.
The first error has been termed ‘The Prosecutor’s Fallacy’. Starkly, it is equating the chance of Innocence, given a DNA match, to the chance of a DNA match, given Innocence. This is logical nonsense: the chance of zero arising, given a fair roulette wheel, is not the same as the chance that the wheel is fair, given that zero occurred. This trap can be avoided by giving the jury an estimate of how many citizens might match the crime scene DNA. With a population of around 60 million, if the match chance is one in 2 million, there might be thirty or so; if it is 1 in 20 million, there might be about three; it is unlikely there are more than half a dozen. But do not overlook the phrase ‘selected at random’: the more close relatives the criminal has, the more matches we would expect, the less strong this evidence against the accused.