n 3, Bayes' theorem states: Using the formula for our example in equation 4: The quantity such as P(H|C) is known as a conditional probability. That is, the conditional probability of H occurring given the evidence C. CHAPTER V BAYES AND THE LAWBayes theorem is used in mathematics and, as I previously mentioned, in many professions. Amongst them is the practice of law. There have been instances where lawyers have taken advantage of the lack of mathematical sophistication among judges and juries by deliberately confusing the two conditional probabilities P(G|E), the probability that the defendant is guilty given the evidence, and P(E|G), the conditional probability that the evidence would be found assuming the defendant would be guilty. Intentional misuse of probabilities has been known to occur where scientific evidence such as DNA testing is involved, such as paternity suits and rape and murder cases. In such cases, prosecuting attorneys may provide the court with a figure for P(E), the probability that the evidence could be found among the general population, whereas the figure of relevance in deciding guilt is P(G|E). As Bayes' formula shows, the two values can be very different, with P(G|E) generally much lower than P(E). Unless there is other evidence that puts the defendant into the group of possible suspects, such use of P(E) is highly suspect. The reason is that, as with the cancer test example, it ignores the initial low prior probability that a person chosen at random is guilty of the crime in question. Instructing the court in the proper use of Bayesian inference was the winning strategy used by American long-distance runner Mary Slaney's lawyers when they succeeded in having her 1996 performance ban overturned. Slaney failed a routine test for performance enhancing steroids at the 1996 Olympic games, resulting in the United States athletic authorities banning her from future competitions. Her lawyers demonstr...
