Exercises

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 ^[1]}

[1] There is a unique Nash equilibrium in the evidence-gathering game presentedin this chapter. Find this equilibrium. Show that in equilibrium, the posteriorprobability is the same as the prior, and so there is no change actual updatingof prior probabilities.