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MIT6_041F10_assn03_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 3 Solutions Due September 29, 2010 1. The hats of n persons are thrown into a box. The persons then pick up their hats at random (i.e., so that every assignment of the hats to the persons is equally likely). What is the probability that (a) every person gets his or her hat back? Answer: n 1 ! . Solution: consider the sample space of all possible hat assignments. It has n ! elements ( n hat selections for the first person, after that n 1 for the second, etc.), with every single- element event equally likely (hence having probability 1 /n !). The question is to calculate the probability of a single-element event, so the answer is 1 /n ! (b) the first m persons who picked hats get their own hats back? ( n m )! Answer: . n ! Solution: consider the same sample space and probability as in the solution of (a). The probability of an event with ( n m )! elements (this is how many ways there are to disribute the remaining n m hats after the first m are assigned to their owners) is ( n m )! /n ! (c) everyone among the first m persons to pick up the hats gets back a hat belonging to one of the last m persons to pick up the hats? Answer: m !( n m )! = n 1 = n 1 . . n ! ( m ) ( n m ) Solution: there are m ! ways to distribute m hats among the first m persons, and ( n m )! ways to distribute the remaining n m hats. The probability of an event with m !( n m )! elements is m !( n m )! /n !. Now assume, in addition, that every hat thrown into the box has probability p of getting dirty (independently of what happens to the other hats or who has dropped or picked it up). What is the probability that (d) the first m persons will pick up clean hats?
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