MIT6_042JF10_rec19_sol

# MIT6_042JF10_rec19_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science November 19, 2010 Tom Leighton and Marten van Dijk Notes 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. Solution. We reason as follows: Pr { A ∩ B } = Pr { A ∩ B } Pr { A ∩ B } Pr { B } = Pr { A ∩ B } Pr { A } Pr { B } · Pr { A } · Pr { A | B } · Pr { B } = Pr { B | A } · Pr { A } In the first step, we rewrite both sides using the facts that Pr { A } and Pr { B } are nonzero. The second step uses the definition of conditional probability. 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. Solution. Pr { rains } Pr { rains | carries umbrella } = Pr { carries umbrella | rains } · Pr { carries umbrella } . 30 = 0 . 80 · . 40 = 0 . 60 We’ve turned around cause and effect! Risk of rain has the effect of making the weatherman carry his umbrella. Yet we’ve shown that if he carries his umbrella, it is pretty likely to rain! 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...
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MIT6_042JF10_rec19_sol - 6.042/18.062J Mathematics for...

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