In an earlier puzzle, you were handed two envelopes, one of which contained twice as much money as the other. After opening one, you were given the chance to swap. At first glance, it appeared that the your chance of getting more money could only increase each time the envelopes were swapped, but clearly this was nonsense: since there is no probability distribution which allows all real numbers to have the same probability, some values would have to have been more likely than others.

Suppose instead envelopes contain the non-negative integer sums 2ⁿ and 2ⁿ⁺¹ with probability q(1 − q)ⁿ for some fixed q < 1/2

Now of course if the envelope you open contains a 1, you know the other must contain 2, so you ought to swap.

But you can do even better than this. Suppose you open an envelope and find an amount of money 2^k

What would the expected value of the second envelope be?

Does this lead to the same paradox?

Given that the first envelope (e₁) has 2^k, the expectation for the second envelope is

E(e₂ | e₁=2^k)
= P(e₂=2^k-1 | e₁=2^k) * 2^k-1 + P(e₂=2^k+1 | e₁=2^k) * 2^k+1

simplifying, we get

E = [P(2^k-1, 2^k) * 2^k-1 + P(2^k, 2^k+1) * 2^k+1] /
[P(2^k-1, 2^k) + P(2^k, 2^k+1)]

= [2^k-1q(1-q)^k-1 + 2^k+1q(1-q)^k] / [q(1-q)^k-1 + q(1-q)^k]

= 2^k-1 * [1 + 4(1-q)] / [1 + (1-q)]

= 2^k-1 * [5 - 4q] / [2 - q]

expanding the fraction, we get

E = 2^k + 2^k-1* [1 - 2q] / [2 - q]

since q < 1/2, 1 - 2q and 2 - q are both positive. Thus,

E > 2^k = e₁.

This may seem like a paradox, but just because the odds are in your favor doesn't mean you'll always win. In fact, we know

2 - q > 1 - 2q,

so E(e₂ | e₁ = 2^k) is between 2^k-1 and 2^k+1, as expected.

Also, note that before either envelope is opened, the expectation for the lesser of the two envelopes is

E(e_low) = 2⁰q(1-q)⁰ + 2¹q(1-q)¹ + 2²q(1-q)² + ...

= q * [(2-2q))⁰ + (2-2q))¹ + (2-2q))² + ...]

and since 2 - 2q > 1, this sum diverges, and the expectation is infinite. This is still not a paradox, but I would sure like to play this game using real money ;).

Edited on August 12, 2005, 12:05 am

Edited on August 12, 2005, 12:06 am

Posted by Josh70679 on 2005-08-11 23:56:08