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 ofthis problem, but when the game is played the envelopes have no markings on them), and (withoutyou or your opponent seeing what she does) she puts $m in envelope 1 and $2 m in envelope 2 forsome m > 0 (treat m as continuous in this problem even though in practice it would have to berounded to the nearest dollar or penny). You and your opponent each get one of the envelopesat random. You open your envelope secretly and find $x (your opponent also looks secretly in hisenvelope), and the referee then asks you if you want to trade envelopes with your opponent. Youreason that if you trade, you will get either $ x2 or $2 x, each with probability 12 . This makes theexpected value of the amount of money you’ll get if you trade equal to12$x2+12($2 x) = $5×4 ,which is greater than the $x you currently have, so you o↵er to trade. The paradox is that youropponent is capable of making exactly the same calculation. How can the trade be advantageousfor both of you?

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