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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?


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.   

Uri now heads the Center for Connected Learning at Northwestern 

Here's a paper by Uri, based on students' reactions to another, 
related paradox:   

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?   

   The St. Petersburg Two-Envelope Paradox   

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   
Associated Topics:
College Probability
High School Probability

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