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hw7 - i th name with probability 1 i At the end choose...

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Math 55 - Homework 7 Due Wednesday, July 27, 2011 Directions: You are encouraged to discuss homework with fellow students. However, you should write up all solutions alone . Your answer should be clear enough that it explains to someone who does not already understand the answer why it works. (This is different than just convincing the grader that you understand.) 1. From the book, do problem 6.4.25 2. From the book, do Chapter 6 Supplemental Excercise 20 3. From the book, do problem 6.2.39 4. You have a very long list of names. You pick one of the names in the following way: Start by remembering the first name. Then, replace the first name with the second name with probability 1 2 . Then, replace the current name with the third name with probability 1 3 . Keep going – at each step, replace the current name with the
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Unformatted text preview: i th name with probability 1 i . At the end, choose whichever name you were remembering. Prove that with this process, all the names on the list are equally likely to be chosen. 5. A bunch of people each pick a number from 1 to n . Then they take the sum of all the numbers mod n . Prove that all outcomes are equally likely if one person picks a number uniformly at random (so all are equally likely) and independent of the other players choices. 6. Prove that there is no way to load two dice so each possible sum , from 2 to 12, is equally likely. (By load the dice, we mean set up the probabilities for each number on each die. The dice remain independent.)...
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