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Section 7.6 Bayes’ Theorem Bayes’ Theorem Let A1, A2, …, An be a partition of a sample space S and let E be an event of the experiment such that P(E) is not zero. Then the posteriori probability P(Ai|E) is given by P Ai | E P Ai P E | Ai P A1 P E | A1 P A2 P E | A2 ... P An P E | An Where 1 i n Posteriori probability: probability is calculated after the outcomes of the experiment have occurred. Ex. A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and 70% of the bulbs respectively. It has been determined that company A’s bulbs are 1% defective while company B’s are 3% defective and company C’s are 4% defective. If a bulb is selected at random and found to be defective, what is the probability that it came from supplier B? Let D = defective P B | D P B P D | B P A P D | A P B P D | B P C P D | C 0.2 0.03 0.1 0.01 0.2 0.03 0.7 0.04 So about 0.17 0.1714