MIT6_042JS10_lec33_sol

# MIT6_042JS10_lec33_sol - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science April 28 Prof. Albert R. Meyer revised April 26, 2010, 1303 minutes Solutions to In-Class Problems Week 12, Wed. Problem 1. [A Baseball Series] The New York Yankees and the Boston Red Sox are playing a two-out-of-three series. (In other words, they play until one team has won two games. Then that team is declared the overall winner and the series ends.) Assume that the Red Sox win each game with probability 3 / 5 , regardless of the outcomes of previous games. Answer the questions below using the four step method. You can use the same tree diagram for all three problems. (a) What is the probability that a total of 3 games are played? (b) What is the probability that the winner of the series loses the first game? (c) What is the probability that the correct team wins the series? Solution. A tree diagram is worked out below. 1st game winner 2nd game winner 3rd game winner 3 games played? winner lost first game? correct team wins? Y R 3/5 2/5 2/5 3/5 R 3/5 Y 2/5 B 3/5 Y 2/5 Y R 3/5 R Y 2/5 outcome YY YRY YRR RYY RYR RR Probability X X X X X X 12/125 18/125 X 9/25 X X 12/125 4/25 18/125 From the tree diagram, we get: 12 18 12 18 12 Pr { 3 games played } = + + + = 125 125 125 125 25 18 12 6 Pr { winner lost first game } = + = 125 125 25 18 18 9 81 Pr { correct team wins } = + + = 125 125 25 125 Creative Commons 2010, Prof. Albert R. Meyer . 2 Solutions to In-Class Problems Week 12, Wed. Problem 2. To determine which of two people gets a prize, a coin is ﬂipped twice. If the ﬂips are a Head and then a Tail, the first player wins. If the ﬂips are a Tail and then a Head, the second player wins. However, if both coins land the same way, the ﬂips don’t count and whole the process starts over. Assume that on each ﬂip, a Head comes up with probability p , regardless of what happened on other ﬂips. Use the four step method to find a simple formula for the probability that the first player wins. What is the probability that neither player wins?...
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## This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec33_sol - Massachusetts Institute of...

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