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A solution and a new problem is posted every Monday evening during fall and spring semesters at www.cs.colostate.edu/~rmm/mathChallenge
One winner each week is eligible for a free ice cream and topping, courtesy of Cold Stone Creamery.
Email your solutions to solution@math.colostate.edu. Indicate your status (undergrad/grad/faculty/other) and school affiliation or city of residence if you are not affiliated with a school.
Email ideas for future challenges to Ross McConnell (rmm@cs.colostate.edu)
Recall that Challenge 10 is a continuation of Challenge 9...
Challenge 9: A game-show host shows you two boxes and tells you that one of them contains x dollars the other contains 2x dollars. He won't tell you the value of x, but he allows you to draw one of the boxes randomly. You have drawn the box containing x dollars with a probability of 1/2.
After opening it, you discover that it has $200 in it. The host now gives you a chance to pocket the money, or give it up in exchange for whatever money is in the other box, which you now know is either $100 or $400. You want to maximize the expected value of your winnings. Should you take him up on the deal? If so, what is your expected gain in doing so?
Challenge 10: For Challenge 9, we got contradictory answers from two camps. Therefore, one of or both of them made illegal steps in their proofs. Challenge 10 is to spot the illegal steps:
Camp 1 claims that it pays to switch: You have drawn the box with the smaller amount of money with a probability of 1/2. If you picked the box with the smaller amount of money, the other box has $400, and if you picked the box with the larger amount, the other box has $100. The expected value of the money in the other box is (1/2)($400) + (1/2)(100) = $250. This is more than the amount y in the box you've picked.
This camp consists of Scott Lundberg, Tim Ellis (undergrad, CSU) Scott Danford, Nathan Behrens (undergrad, Colorado School of Mines), Gabriel Somlo, Matt Gibbs, Kyriakos Chatzidimitriou, Cliff Jones (grad, CSU), Andrew Johnson (grad, Colorado School of Mines), Florian Hulpke (grad, University of Hannover), Robert St. John (CSU), Byung-Soo Kim (South Korea), Nicolae Popescu (Fort Collins), Mike Darschewski (Denver).
Camp 2 claims that Camp 1 is wrong. This camp consists only of Ian Ellis and Saravanan Sellappa. The two of them have given slightly different arguments; I will paraphrase Ian's.
If you pick the box with x dollars, then you will gain x by switching, and if you picked the box with 2x dollars, you will lose x by switching. The expected gain from switching is (1/2)(-x) + (1/2)x = 0. There's no reason to switch.