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Ch. 6 Bayes' Rule

Ch. 6 Bayes' Rule - 1 Inductive Logic PHIL 111 Fall 2009...

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1 Inductive Logic PHIL 111 Fall 2009 Chapter 7 Bayes’ Rule Bayes’ Rule: Pr(H/E) = Pr(H j ) Pr(E/H j )]) / ∑ [Pr(H i) Pr(E/H i )] = Pr(H)Pr(E/H) / Pr(H)Pr(E/H) + Pr( H)Pr(E/ H) 1 . A tarantula is a large, fierce-looking, and somewhat poisonous tropical spider. Once upon a time, 3% of consignments of bananas from Honduras were found to have tarantulas on them, and 6% of the consignments from Guatemala had tarantulas. 40% of the consignments came from Honduras and 60% came from Guatemala. Let G = The lot came from Guatemala. Let H = The lot came from Honduras. Let T = The lot had a tarantula on it. a. A tarantula was found on a randomly selected lot of bananas. What is the probability that this lot came from Guatemala? b. What is the probability that this lot came from Honduras, given that a tarantula was discovered in a randomly selected lot of bananas? c. What is the probability that a lot had tarantulas in it, given that it came from Honduras? 2 . [Base rates] Odd Question 5 You have been called to jury duty in a town where there are two taxi companies, Green Cabs Ltd. and Blue Taxi Inc. Blue Taxi uses cars painted blue. Green Cabs uses green cars. Green Cabs dominate the market, with 85% of the taxis on the road. On a misty winter night, a taxi side swiped another car and drove off. A witness says it was blue cab. The witness is tested under conditions like those on the night of the accident, and 80% of the time she correctly reports the color of the cab that is seen. That is, regardless of whether she is shown a blue or a green cab in misty evening light she gets the color right 80% of the time. Let G = A taxi selected at random is green. Let B = A taxi selected at random is blue. Let W b = A witness claims that the taxi is blue. a. The witness says it was a blue cab. What is the probability that it was a blue cab? b. Mark the branch of the probability tree with a (1) indicating the true positive result, with a (2) indicating the false positive result, with a (3) indicating the true negative result, and with a (4) highlighting the false negative result of the test. 3 . Consider a pregnancy test that claims to detect 99% of all pregnancies it is used on; it will produce a false positive report 5% of the time. a. Suppose that to begin with there is a 99% chance that a woman is not pregnant – because she has been using a method of birth control which claims to be 99% effective. Suppose she uses this pregnancy test and it says that she is pregnant, i.e. the test says “yes” (Y). What is the probability that she is pregnant given this result?

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