Closed Envelopes Paradox
Date: 09/13/2001 at 02:43:38 From: dror Subject: The closed envelopes paradox Dear Dr. Math, Can you please explain the following paradox: You have two closed envelopes. One of them contains a certain amount of money and the other one contains twice as much. You have to choose only one of them. Suppose you want to choose the first one. If that envelope contains, say, $1000, the other one may contain $2000 or $500. Namely, if you want to switch your choice now, you can win $1000 or lose $500. On the average, you will win $250. Therefore, it seems always worthwhile to exchange your choice. Why then not to choose the second envelope in the first place? Because, by the same consideration, it would be always worthwhile to switch your choice to the first one again - and so on. Every envelope you choose - the second one is preferable on the average, and that conclusion is obviously absurd. Where did this paradox come from? Thanks!
Date: 09/13/2001 at 06:25:42 From: Doctor Mitteldorf Subject: Re: The closed envelopes paradox The paradox was authored by Uri Wilensky when he was a thesis student at MIT's Media Lab, ca 1990. http://www.media.mit.edu/people/uriw/ Uri now heads the Center for Connected Learning at Northwestern University: http://www.ccl.sesp.northwestern.edu/ Here's a paper by Uri, based on students' reactions to another, related paradox: http://ccl.northwestern.edu/papers/paradox/lppp/ After reading this, you'll see why I'm reluctant to just tell you the answer. Confronting these paradoxes is a great learning tool, especially if it is done with guidance. The reason for the paradox is deep - it's not just a trick where you'll say, "oh yeah - why didn't I think of that?" In fact, you can't "solve" the paradox without substantially broadening your appreciation of what probability calculations are and are not. My explicating the answer would short- circuit that whole process, and you'd get a lot less out of it - but I'd love to initiate a correspondence with you, where we work on the implications together. Reading the above paper is one good place to start. Here's another: Try working through some examples. You can actually make the envelopes and try it, or imagine in detail the process of actually making the envelopes. What would it mean to construct a representative sample of the possible sets of envelopes? If you're comfortable with computer programming, you can actually write a simulation in which many pairs of random envelopes are constructed and tried. Test whether the second envelope really does come out better. Even if you're not a programmer, it will be very enlightening to plan writing that program in great detail. For those readers who are already comfortable thinking in terms of continuous, normalized probability distributions, there is an in-depth analysis of the paradox at The Two-Envelope Paradox: A Complete Analysis? http://www.u.arizona.edu/~chalmers/papers/envelope.html The St. Petersburg Two-Envelope Paradox http://www.u.arizona.edu/~chalmers/papers/stpete.html But if you're still savoring the mystery of this problem, please don't peek! at least until you've learned all that you can from thinking about the paradox yourself and talking to others. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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