hw8 - to midnight, you put in balls 11 through 20 and again...

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Math 55 - Homework 8 Due Tuesday/Wednesday, August 2/3, 2011 Directions: You should work on this homework in groups of three . Your group should sign up for a 40 minute timeslot on Tuesday or Wednesday. Each group member will present an answer to one of three problems, chosen at random. 1. We say that a real number r is algebraic if there is some nonzero polynomial f ( x ) with integer coefficients so that f ( r ) = 0. For example, 3 2 and 3 2 are algebraic because they solve x 3 - 2 = 0 and 2 x - 3 = 0 respectively. ( π and e are not algebraic, but this is hard to prove.) Prove that the set of algebraic numbers is countable. 2. You have a very large urn. At one minute to midnight, you put in balls which are numbered 1 through 10, and then remove one ball chosen at random. At half a minute
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Unformatted text preview: to midnight, you put in balls 11 through 20 and again remove one ball chosen at random. Continuing in this manner, at 1 2 i minutes to midnight, you put balls 10 i + 1 through 10 i + 10 into the urn and remove one ball at random. Prove that for any given ball k , the probability ball k is in the urn at midnight is 0. 3. You are given an unfair coin. The coin comes up heads with some probability p not known to you. The ips are independent and p does not change. Describe a way to use this bad coin to simulate a fair coin. (So you should describe some experiment with the coin and some event in that experiment which occurs with probability exactly 1 2 .)...
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This note was uploaded on 03/14/2012 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.

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