- 1. Suppose that statistically, only 0.5 per cent of individuals are criminals. Suppose that this is the court's prior belief. Thus p(G) = 0.005. Suppose that evidence correctly identifies a guilty defendant 99 per cent of the time. That is, p(E | G) = 0.99. Suppose also that if there is no evidence, this correctly identifies an innocent defendant 99 per cent of the time. Therefore, p(NE | I) = 0.99 This means that p(E | I) = 0.01.
- (a) Use Bayes' rule to find P(G | E), the probability that the defendant is guilty, given that evidence has been presented.
- (b) Given that the piece of evidence has been produced, what is the probability that the defendant is innocent?
- (c) Now suppose that evidence is not presented. What is the probability that the defendant is actually guilty, given that there is no evidence of this?
- (d) What is the probability that he is innocent? In other words, find P(G | NE) and P (G | NE). 2