3. Bayesian methods: a) Derive Bayes' Theorem from the product rule / definition of conditional probability (1 mark). Prob(B | A) = b) For this new well, your x-ray test indicates that a valve on the blow out preventer is cracked. Should you replace it? The test has a 99.9% chance of detecting a crack if present (true positive rate), but a 0.001% chance of detecting a crack if not present (false positive rate). There is a 0.4% chance to have a crack. What is the probability that the valve is cracked given the test says it is (2 marks)? P(BIA) = P(A|B) P(B) = P(A|B) P(B) Recall Bayes' Theorem P(A) P(A|B) P(B) + P(A|B) P(B)