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# ws730 - Rob Bayer Bayes Rule Math 55 Worksheet 1 1 Suppose...

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Rob Bayer Math 55 Worksheet July 30, 2009 Bayes Rule 1. Suppose you have 3 fair dice and 1 loaded die that comes up 6 with probability 1 3 and each other value with probability 2 15 . If you pick a die from this set at random, roll it twice, and get a 6 twice, what’s the probability that you picked the loaded die? 2. A test for a certain disease is 99% speciﬁc (that is, 99% of the time the test will be negative for someone without the disease) and 98% sensitive (ie, 98% of the time the test will be positive for someone who has the disease). (a) If you test positve, what’s the probability that you actually have the disease? (b) If you test positive twice, how likely is it that you have the disease? Assume the test results are independently distributed 3. Prove the following generalization of Bayes’ Rule: If F 1 ,F 2 ,...,F n are disjoint events of positive probability such that F 1 F 2 ∪···∪ F n = S and E is some event with positive probability, then p ( F 1 | E ) = p ( E

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ws730 - Rob Bayer Bayes Rule Math 55 Worksheet 1 1 Suppose...

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