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Date: 09/13/2001 at 02:43:38
From: dror

Dear Dr. Math,

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

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,

After reading this, you'll see why I'm reluctant to just tell you the
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
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.

continuous, normalized probability distributions, there is an in-depth

The Two-Envelope Paradox: A Complete Analysis?
http://www.u.arizona.edu/~chalmers/papers/envelope.html

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

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Probability
High School Probability

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