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Bayes’ Theorem Bayes Theorem is a restatement of the definition of conditional probability combined with the law of total probability. Conditional Probability: If A, B are events in sample space S then by definition P (A|B) = but as P (B|A) = P (A ∩ B) P (B) P (A ∩ B) ⇐⇒ P (A ∩ B) = P (B|A)P (A) this implies from our first equation: P (A) P (B|A)P (A) P (A ∩ B) = P (A|B) = P (B) P (B) Now for the Law of Total Probability. If {Ei }, i = 1, ..., n is a partition of S, that is Ei ∩ Ej = ∅, i 6= j and ∪i Ei = S then for any event B then trivially (B ∩ Ei ) ∩ (B ∩ Ej ) = ∅ i 6= j and we have X P (B) = P (B ∩ (∪i Ei )) = P (B ∩ Ei ). i So using this last equation we have Bayes: P (A|B) = P (B|A)P (A) P (B|A)P (A) P (A ∩ B) = =P P (B) P (B) i P (B ∩ Ei ) If A = Ej for some j then we have the usual representation: P (B|Ej )P (Ej ) P (B|Ej )P (Ej ) P (Ej ∩ B) = = P P (Ej |B) = P (B) P (B) i P (B ∩ Ei ) SOA Exam P: Bayes sample problems • 19. An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers: Age of Driver Prob of accident Portion of company’s clients 16-20 0.06 0.08 21-30 0.03 0.15 31.65 0.02 0.49 66-99 0.04 0.28 A randomly selected driver that the company insures has an accident. Calculate the probability that the driver was age 16-20. (Ans: 0.16) • 22. A health study tracked a group of persons for five years. At the beginning of the study 20% were classified as heavy smokers, 30% as light smokers, and 50% as nonsmokers. Results of the study showed that light smokers were twice as likely as nonsmokers to die during the five-year study, but only half as likely as heavy smokers. A randomly selected participant from the study died over the five-year period. Calculate the probability that the participant was a heavy smoker. (Ans: 0.42) • 28. A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X’s shipments 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hopsital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X? (Ans: 0.10)