A Supreme Court considered the use of Bayes Theorem to combine items of evidence in OJ Simpsons case. Before any prosecution or defence evidence is presented, it might be reasonable to assume that the culprit is a male aged between 24 and 59 years old. There were approximately 120 000 males of this age who lived locally to where the crime took place. Of course, the culprit may not be local. This increases the hypothesized number of possible men who could be the culprit from around 120 000 to 160 000, say.

The prosecution case rested on forensic evidence, which will be called E:

E = DNA match between Simpsons and a sample, accepted as being from the culprit, taken from the victim.

The prosecutor's forensic expert testified that for a randomly chosen person, the probability that their DNA matched that of the sample was 1/160 000 000. The defence argued that this probability was in fact 1/1 600 000.

a. What is the nave probability that Simpsons is guilty? (3 marks)

b. The prosecutor's forensic expert testified argued that

P(Simpsons not guilty | E) = 1/160 000 000.

Do you support the definition and interpretation of this conditional probability? Support your argument. (5 marks)

c. The judge and jury require an estimate of the probability that Simpsons is guilty, given the evidence E. Calculate estimate of this probability. (9 marks)

d. Would you recommend use of Bayes Theorem in a similar case? Give reason for your answer.