The Two-Envelope Paradox
Suppose you're on a gameshow where you can choose either of two sealed envelopes, A or B,
both containing money. The host doesn't say how much money is in each, but he does let you know
that one envelope contains twice as much as the other.
You pick envelope A, open it and see that it contains $100. The host then makes the
following offer: you can either keep the $100, or you can trade it for whatever is in envelope
You might reason as follows: since one envelope has twice what the other one has, envelope
B either has 200 dollars or 50 dollars, with equal probability. If you switch, then, you stand
to either win $100 or to lose $50. Since you stand to win more than you stand to lose, you
But just before you tell the host you would like to switch, another thought might occur to
you. If you had picked envelope B, you would have come to exactly the same conclusion. So if the
above argument is valid, you should switch no matter which envelope you choose. But that can't
What's wrong with your reasoning?
Some mathematicians have argued that the problem here has to do with assigning a probability
measure on an infinite set (the natural numbers). But logician Raymond Smullyan pointed out that
the paradox can be restated without involving probability. Smullyan's restatement is as follows
(from Smullyan's Satan, Cantor, and Infinity):
We can prove two contradictory propositions:
"Proposition 1. The amount that you will gain, if you do gain, is greater than the
amount you will lose, if you do lose.
"Proposition 2. The amounts are the same."
The proof of Proposition 1 is essentially the one already given: "Let n be the number of
dollars in the envelope you are now holding. Then the other envelope has either 2n or n/2
"...If you gain on the trade, you will gain n dollars, but if you lose on the trade,
you will lose n/2 dollars. Since n is greater than n/2, then the amount you gain, if you do
gain which is n is greater than the amount you will lose, if you do lose
which is n/2. This proves Proposition 1.
"Now for the proof of Proposition 2. Let d be the difference between the amounts in the two
envelopes, or what is the same thing, let d be the lesser of the two amounts. If you gain on the
trade, you will gain d dollars, and if you lose on the trade, you will lose d dollars. And so
the amounts are the same after all... This proves Proposition 2."
©2000 Franz Kiekeben