MIT6_042JF10_rec18_sol

MIT6_042JF10_rec18_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science November 17, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 18 The Total Probability Law is a handy tool for breaking down the computation of a probability into distinct cases. Theorem 1 (Total Probability Law) . Let E and X be events. Then Pr { E } = Pr { E | X } Pr { X } + Pr E | X Pr X provided < Pr { X } < 1 . Proof. Lets simplify the right side. Pr { E | X } Pr { X } + Pr E | X Pr X = Pr { E X } Pr { X } + Pr E X Pr X Pr { X } Pr X = Pr { E X } + Pr E X = Pr { E } The first step uses the definition of conditional probability. On the next-to-last line, were adding the probabilities of all outcomes in E and X to the probabilities of all outcomes in E and not in X . Since every outcome in E is either in X or not in X , this is the sum of the probabilities of all outcomes in E , which equals Pr { E } by the definition of the probability of an event. The theorem generalizes as follows: Theorem 2. Let E be an event and let X 1 ,...,X n be disjoint events whose union is the entire sample space. Then n Pr { E } = Pr { E | X i } Pr { X i } i =1 provided < Pr { X i } < 1 . Recitation 18 2 1 Nerditosis There is a rare and deadly disease called Nerditosis which afflicts about 1 person in 1000. One symptom is a compulsion to refer to everything fields of study, classes, buildings, etc. using numbers. Its horrible. As victims enter their final, downward spiral, theyre awarded a degree from MIT. Two doctors claim that they can diagnose Nerditosis....
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MIT6_042JF10_rec18_sol - 6.042/18.062J Mathematics for...

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