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MIT6_042JF10_rec19

# MIT6_042JF10_rec19 - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science November 19, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 19 1 Bayes’ Rule Bayes’ Rule says that if A and B are events with nonzero probabilities, then: Pr { A | B } · Pr { B } = Pr { B | A } · Pr { A } a. Prove Bayes’ Rule. b. A weatherman walks to work each day. Some days it rains: Pr { rains } = 0 . 30 Sometimes the weatherman brings his umbrella. Usually this is because he predicts rain, but he also sometimes carries it to ward off bright sunshine. Pr { carries umbrella } = 0 . 40 As a weatherman, he usually doesn’t get caught out in a storm without protection: Pr { carries umbrella | rains } = 0 . 80 Suppose you see the weatherman walking to work, carrying an umbrella. What is the probability of rain? Use Bayes’ Rule.

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Recitation 19 2 2 DNA Profiles Suppose that we create a a national database of DNA profiles. Let’s make some (overly) simplistic assumptions: Each person can be classified into one of 20 billion different “DNA types”. (For ex- ample, you might be type #13,646,572,661 and the person next to
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