MIT6_042JS10_lec36_sol

MIT6_042JS10_lec36_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science May 5 Prof. Albert R. Meyer revised May 5, 2010, 857 minutes Solutions to In-Class Problems Week 13, Wed. Problem 1. Let’s see what it takes to make Carnival Dice fair. Here’s the game with payoff parameter k : make three independent rolls of a fair die. If you roll a six • no times, then you lose 1 dollar. • exactly once, then you win 1 dollar. • exactly twice, then you win two dollars. • all three times, then you win k dollars. For what value of k is this game fair? Solution. Let the random variable P be your payoff. Then we can compute E [ P ] as follows: E [ P ] = 1 Pr { 0 sixes } + 1 Pr { 1 six } + 2 Pr { 2 sixes } + k Pr { 3 sixes } · · · · ± 3 ± ± 2 ± 2 ± ± 3 5 1 5 1 5 1 = 1 · 6 + 1 · 3 6 6 + 2 · 3 6 6 + k · 6 125 + 75 + 30 + k = 216 The game is fair when E [ P ] = 0 . This happens when k = 20 . Problem 2. A classroom has sixteen desks arranged as shown below. Creative Commons 2010, Prof. Albert R. Meyer .

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± ² ² ² ± 2 Solutions to In-Class Problems Week 13, Wed. If there is a girl in front, behind, to the left, or to the right of a boy, then the two of them ﬂirt . One student may be in multiple ﬂirting couples; for example, a student in a corner of the class- room can ﬂirt with up to two others, while a student in the center can ﬂirt with as many as four others. Suppose that desks are occupied by boys and girls with equal probability and mutually
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MIT6_042JS10_lec36_sol - Massachusetts Institute of...

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