A Bayesian Puzzler
                                     
 The two envelope problem
 
 Here's an seemingly straightforward probability puzzle that defies
 intuition. You are offered a choice between two envelopes. Each
 envelope contains an unstated amount of cash and you are told that one
 of the envelopes contains twice as much money as the other. You pick
 one of them and look inside and find ten dollars. That means the other
 envelope has either $5 or $20.
 
 Now you are offered the choice of keeping what you have or exchanging
 it for the other (unopened) envelope.
 
 should you switch?
 
 Naively, one thinks "well, the odds that I picked the envelope with
 the most money were 50-50 at the start and nothing has changed. It
 makes no difference that I looked inside of this one, thus my odds are
 still 50-50 and It makes no difference whether I keep what I have or
 switch. On the averege I can't do any better".
 
 on the other hand , One might consider the 'expected value' of
 switching: "There is a 50-50 chance that the envelope contains $20 and
 a 50-50 chance that it contains $5. If we repeated this game many
 times and I switched when I saw $10 in my envelope then half the time
 I wind up with $5 and half the time I would wind up with $20. Thus on
 the average the other envelope will be worth $12.50.
 that is: (0.5)*($5) + (0.5)*($20) = $12.50 
 Since it will cost me $10 to make the switch (you must give up the
 envelope you have), the expected net gain for switching is $2.50. So
 it makes sense to switch!"
 
 or does it? Do either of these two arguments make sense?
 
 A further twist
 
 Suppose there had been another person in the game. This person gets
 the envelope you don't choose. You both look inside your envelopes but
 dont tell the other person how much you got. Then you both are offered
 the opportunity to exchange envelopes. And by the above logic you both
 want to switch! Worse yet you both would be willing to pay for the
 privilege of switching: in your case, seeing $10, you would be willing
 to pay anything less that $2.50 for the privilege of switching. Is
 this possible? After all you had complete free will over your choice
 at the start.
 
 The ultimate twist
 
 Actually its a lot worse than that. Suppose you don't look in the
 envelope you picked. You know there is some unknown amount in the
 envelope, call it X. The other envelope has either 2X or 0.5X. The
 expected value of the other envelope is:
 0.5*(0.5X) + 0.5*(2X) = 1.25X
 Thus if you switch you expect to gain 0.25X. So which ever envelope
 you choose to start with, you immediately want to switch it before you
 even look at it. And of course once you get that envelope you want to
                  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 switch back.....
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