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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science April 14, 2005 Srini Devadas and Eric Lehman Problem Set 9 Solutions Due: Monday, April 25 at 9 PM Problem 1. There are three coins: a penny, a nickel, and a quarter. When these coins are ﬂipped: • The penny comes up heads with probability 1 / 3 and tails with probability 2 / 3 . • The nickel comes up heads with probability 3 / 4 and tails with probability 1 / 4 . • The quarter comes up heads with probability 3 / 5 and tails with probability 2 / 5 . Assume that the way one coin lands is unaffected by the way the other coins land. The goal of this problem is to determine the probability that an odd number of coins come up heads. For this first problem, we’ll closely follow the fourstep procedure for solving probability problems described in lecture. Your solution should include a tree diagram. (a) What is the sample space for this experiment? Solution. We can regard each outcome as a triple indicating the orientation of the penny, nickel, and quarter. For example, the triple ( H, T, H ) is the outcome in which the penny is heads, the nickel is tails, and the quarter is heads. The sample space is the set of all such triples: { H, T } 3 . (b) What subset of the sample space is the event that an odd number of coins come up heads? Solution. The event that an odd number of coins come up heads is the subset: { ( H, H, H ) , ( H, T, T ) , ( T, H, T ) , ( T, T, H ) } (c) What is the probability of each outcome in the sample space? Solution. Edges in the tree diagram are labeled with the probabilities given in the problem statement. The probability of each outcome is the product of the probabil ities along the corresponding roottoleaf path. The resulting outcome probabilities are noted in the tree diagram. 2 Problem Set 9 X X X X X X X X X X X X X X X X X X X X X X X X H H H H H H H H H H H H @ @ @ @ @ @ H T H T H T T H T H T H T H penny nickel quarter odd? × × × × prob. 1 / 3 2 / 3 3 / 4 1 / 4 3 / 4 1 / 4 3 / 5 2 / 5 3 / 5 2 / 5 3 / 5 2 / 5 3 / 5 2 / 5 9 / 60 6 / 60 3 / 60 2 / 60 18 / 60 12 / 60 6 / 60 4 / 60 (d) What is the probability that an odd number of coins come up heads? Solution. The probability of an event is the sum of the probabilities of the outcomes in that event. In this case: Pr ( odd number of heads ) = Pr ( { ( H, H, H ) , ( H, T, T ) , ( T, H, T ) , ( T, T, H ) } ) = Pr (( H, H, H )) + Pr (( H, T, T )) + Pr (( T, H, T )) + Pr (( T, T, H )) = 9 60 + 2 60 + 12 60 + 6 60 = 29 60 Problem 2. 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 Mustard. Otherwise, Colonel Mustard escapes. Exactly one of the three has an opportunity with the following probabilities: Pr ( Plum has opportunity ) = 1 / 6 Pr ( Green has opportunity ) = 2 / 6 Pr ( Scarlet has opportunity ) = 3 / 6 Exactly one has the revolver with the following probabilities, regardless of who has an...
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 Spring '11
 Dr.EricLehman
 Computer Science, Probability, Sauron, Colonel Mustard

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