Exercise 1. (Probabilities and Bayes Theorem) [10 points]

All cars are either red or blue. The witness claimed the car that hit the pedestrian was blue. Witnesses are believed to be about 80% reliable in reporting car color (regardless of the actual car color). But only 10% of all cars are blue.

(a) Write an equation relating the following quantities and perhaps other quantities:

p(true = blue) = p(T=b) p(true = blue | claimed = blue) = p(T=b|C=b) p(claimed = blue | true = blue) = p(C=b|T=b)

(b) Match the three probabilities above with the following terms: prior probability, probability of the data given the evidence, posterior probability.

(c) Give the values of all three probabilities. (Hint: Use Bayes Theorem.) Which probability should the judge care about?