Two Envelopes Problem - The Problem

The Problem

The basic setup: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are offered the possibility to take the other envelope instead.

The switching argument: Now suppose you reason as follows:

  1. I denote by A the amount in my selected envelope.
  2. The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
  3. The other envelope may contain either 2A or A/2.
  4. If A is the smaller amount, then the other envelope contains 2A.
  5. If A is the larger amount, then the other envelope contains A/2.
  6. Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.
  7. So the expected value of the money in the other envelope is



  8. This is greater than A, so I gain on average by swapping.
  9. After the switch, I can denote that content by B and reason in exactly the same manner as above.
  10. I will conclude that the most rational thing to do is to swap back again.
  11. To be rational, I will thus end up swapping envelopes indefinitely.
  12. As it seems more rational to open just any envelope than to swap indefinitely, we have a contradiction.

The puzzle: The puzzle is to find the flaw in the very compelling line of reasoning above.

Read more about this topic:  Two Envelopes Problem

Famous quotes containing the word problem:

    “We have heard all of our lives how, after the Civil War was over, the South went back to straighten itself out and make a living again. It was for many years a voiceless part of the government. The balance of power moved away from it—to the north and the east. The problems of the north and the east became the big problem of the country and nobody paid much attention to the economic unbalance the South had left as its only choice.”
    —Lyndon Baines Johnson (1908–1973)

    “[How] the young . . . can grow from the primitive to the civilized, from emotional anarchy to the disciplined freedom of maturity without losing the joy of spontaneity and the peace of self-honesty is a problem of education that no school and no culture have ever solved.”
    —Leontine Young (20th century)