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Unformatted text preview: Envelopes * L A T E X file: EnvelopesPuzzle — Daniel A. Graham <[email protected]>, June 16, 2005 An honest but mischievous father tells his two riskneutral sons, Andy and Bill, that he has placed 10 n dollars in one envelope and 10 n + 1 dollars in another envelope, where n was chosen with equal probability among the integers between 1 and 6. The sons completely believe their father. He then randomly hands each son one of the envelopes and the sons privately open their envelopes. Andy finds 10 4 = 10,000 dollars in his envelope and concludes that the other envelope contains either $1,000 or $100,000 with equal probability and thus has an expected value of $50,500. Bill finds 10 3 = 1,000 dollars in his envelope and concludes that the other envelope has $100 or $10,000 with equal probability and thus has an expected value of $5,500. Case A. Suppose the father privately asks each son whether or not he would be willing to pay $1 to switch envelopes with the other son. Both sons say yes. Suppose the father reports this to each son and repeats the question. Both sons again say yes, the father reports this fact and repeats the question a third time. Again both say yes but if the father reports this and asks a fourth time, then Bill will saya third time....
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 Spring '09
 Physics, Probability theory, Andy, Andy ﬁnds

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