Rest of the problems are in the attached file. I also attached book.
1.(the Exchange Paradox) You’re playing the following game against an opponent, with a referee also taking part. The referee has two envelopes (numbered 1 and 2 for the sake of this problem, but when the game is played the envelopes have no markings on them), and (without you or your opponent seeing what she does) she puts $m in envelope 1 and $2 m in envelope 2 for some m > 0 (treat m as continuous in this problem even though in practice it would have to be rounded to the nearest dollar or penny). You and your opponent each get one of the envelopes at random. You open your envelope secretly and find $x (your opponent also looks secretly in his envelope), and the referee then asks you if you want to trade envelopes with your opponent. You reason that if you trade, you will get either $ x 2 or $2 x, each with probability 1 2 . This makes the expected value of the amount of money you’ll get if you trade equal to 1 2 $x 2 + 1 2 ($2 x) = $5x 4 , which is greater than the $x you currently have, so you o↵er to trade. The paradox is that your opponent is capable of making exactly the same calculation. How can the trade be advantageous for both of you?
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