3. Bayesian methods: a] Derive Bayes' Theorem from the product rule 1 definition of conditional probability [1 mark}. Pron | A) = Pm n B) PCAIB) PfCB) = PCBIA) PGDl 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 030196 chance of detecting a crack if not present {false positive rate]. There is a {1.4% chance to have a crack. Ill'u'hat is the probability that the valve is cracked given the test says it is not {2 marks]