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MIT6_042JS10_lec36_prob

MIT6_042JS10_lec36_prob - 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 4, 2010, 1304 minutes 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? Problem 2. A classroom has sixteen desks arranged as shown below. 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 independently. What is the expected number of ﬂirting couples? Hint: Linearity.

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