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MIT6_042JF10_assn11

# MIT6_042JF10_assn11 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science November 16, 2010 Tom Leighton and Marten van Dijk Problem Set 11 Problem 1. [20 points] You are organizing a neighborhood census and instruct your census takers to knock on doors and note the sex of any child that answers the knock. Assume that there are two children in a household and that girls and boys are equally likely to be children and to open the door. A sample space for this experiment has outcomes that are triples whose Frst element is either B or G for the sex of the elder child, likewise for the second element and the sex of the younger child, and whose third coordinate is E or Y indicating whether the e lder child or y ounger child opened the door. ±or example, ( B , G , Y ) is the outcome that the elder child is a boy, the younger child is a girl, and the girl opened the door. (a) [5 pts] Let T be the event that the household has two girls, and O be the event that a girl opened the door. List the outcomes in T and O . (b) [5 pts] What is the probability Pr ( T | O ), that both children are girls, given that a girl opened the door? (c) [10 pts] Where is the mistake in the following argument for computing Pr ( T | O )? If a girl opens the door, then we know that there is at least one girl in the household. The probability that there is at least one girl is 1 Pr (both children are boys) = 1 (1 / 2 × 1 / 2) = 3 / 4 . So, Pr ( T | there is at least one girl in the household) Pr ( T there is at least one girl in the household) = Pr { there is at least one girl in the household } Pr ( T ) = Pr { there is at least one girl in the household } = (1 / 4) / (3 / 4) = 1 / 3 . Therefore, given that a girl opened the door, the probability that there are two girls in the household is 1/3.

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2 Problem Set 11 Problem 2. [20 points] Professor Plum, Mr. Green, and Miss Scarlet are all plotting to shoot Colonel Mustard. If one of these three has both an opportunity and the revolver , then that person shoots Colonel
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MIT6_042JF10_assn11 - 6.042/18.062J Mathematics for...

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