MIT6_042JF10_rec21

MIT6_042JF10_rec21 - up on both? (b) [ pts] What is the...

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6.042/18.062J Mathematics for Computer Science December 1, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 21 Problem 1. [ points] Here’s yet another fun 6.042 game! You pick a number between 1 and 6. Then you roll three fair, independent dice. If your number never comes up, then you lose a dollar. If your number comes up once, then you win a dollar. If your number comes up twice, then you win two dollars. If your number comes up three times, you win four dollars! What is your expected payoff? Is playing this game likely to be proFtable for you or not?
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2 Recitation 21 Problem 2. [ points] The number of squares that a piece advances in one turn of the game Monopoly is determined as follows: Roll two dice, take the sum of the numbers that come up, and advance that number of squares. If you roll doubles (that is, the same number comes up on both dice), then you roll a second time, take the sum, and advance that number of additional squares. If you roll doubles a second time, then you roll a third time, take the sum, and advance that number of additional squares. However, as a special case, if you roll doubles a third time, then you go to jail. Regard this as advancing zero squares overall for the turn. (a) [ pts] What is the expected sum of two dice, given that the same number comes
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Unformatted text preview: up on both? (b) [ pts] What is the expected sum of two dice, given that dierent numbers come up? (Use your previous answer and the Total Expectation Theorem.) 3 Recitation 21 (c) [ pts] To simplify the analysis, suppose that we always roll the dice three times, but may ignore the second or third rolls if we didnt previously get doubles. Let the random variable X i be the sum of the dice on the i-th roll, and let E i be the event that the i-th roll is doubles. Write the expected number of squares a piece advances in these terms. (d) [ pts] What is the expected number of squares that a piece advances in Monopoly? MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.

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MIT6_042JF10_rec21 - up on both? (b) [ pts] What is the...

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