# pset8 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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EECS 310, Fall 2011 Instructor: Nicole Immorlica Problem Set #8 Due: November 22, 2011 1. (30 points) Let S = { 1 ,...,n } and let π be a random permutation on S (i.e., π : S S is a random bijection). Let π ( k ) represent the k th element in the permutation π . For a subset A of S let f ( A ) be the minimum of π ( a ) over all elements of a A . For example, if n = 10, A = { 2 , 5 , 7 } , and π (2) = 3 (5) = 10 (7) = 2, then f ( A ) = 2. Let A and B be two arbitrary subsets of S . Express the following in terms of A , B , and S . Be sure to explain your calculations. (a) (6 points) What is the probability that π (1) = 1? (b) (12 points) For an arbitrary element a A , what’s the probability that f ( A ) = π ( a )? (c) (12 points) What is the probability f ( A ) = f ( B )? 2. (25 points) Consider the following hat puzzle: Three players enter a room wearing a hat. Each hat is either red or blue, and the color of a hat is chosen uniformly at random. A player can see every hat but his own. Each player must write on a card either red , blue , or pass . After everyone writes down something, the cards are simultaneously revealed (so no helpful information is revealed to the other players by what is written down on

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pset8 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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