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I don't see the paradox... If you have 2x and swap you lose x. If you have x and swap, you gain x. 0.5(-x) + 0.5(x) = 0, So you should be indifferent to swap

I don't see the paradox... If you have 2x and swap you lose x. If you have x and swap, you gain x. 0.5(-x) + 0.5(x) = 0, So you should be indifferent to swapping. A I missing something? Not to say I don't make mistakes, but I have a BS in mathematics so maybe this is only obvious for people with a background in math? EDIT: No need for dollar values.

oneeyedpigeon 13y ago
I don't have a BS in maths, but the important bit, as I read it, is that you don't know the values involved. I.e. if you have 20, there is either 10 in the other envelope, or 40. You have no way of knowing which is the case, so it's in your interest to swap since the benefits outweigh the risks. This is totally counterintuitive, though, so I'm fully willing to accept I'm missing something! And I don't buy the 'indefinite swapping' argument, since the second swap must surely reverse any advantage gained.
- andrewflnr 13y ago
  Once you have the second envelope in hand, since you don't actually know what's in it, the exact same expected-value argument applies to switching back: there might be $10 or $40 in the other (first) envelope, "so it's in your interest to swap since the benefits outweigh the risk". While the second swap would reverse any advantage possibly gained, it would also reverse any possible harm sustained You dont know. The situation is perfectly symmetric; picking and switching is the same as just picking the second one at first. So just pick one.
  - colomon 13y ago
    Env A has $20 (we have it and know). Env B has $10 or $40 (equal chance). We swap. Now we hold Env B, which has $10 or $40 in it. We know the other envelope (Env A) is $20. Why would we switch again?
    - andrewflnr 13y ago
      The original statement of the problem says you're offered the choice before you open the envelope. I guess I was sloppy about the $20 assumption too. As for the case you do know, I still feel wrong about the conclusion that you should switch, but I can't formalize it.

oneeyedpigeon 13y ago

I don't have a BS in maths, but the important bit, as I read it, is that you don't know the values involved. I.e. if you have 20, there is either 10 in the other envelope, or 40. You have no way of knowing which is the case, so it's in your interest to swap since the benefits outweigh the risks. This is totally counterintuitive, though, so I'm fully willing to accept I'm missing something! And I don't buy the 'indefinite swapping' argument, since the second swap must surely reverse any advantage gained.

andrewflnr 13y ago
Once you have the second envelope in hand, since you don't actually know what's in it, the exact same expected-value argument applies to switching back: there might be $10 or $40 in the other (first) envelope, "so it's in your interest to swap since the benefits outweigh the risk". While the second swap would reverse any advantage possibly gained, it would also reverse any possible harm sustained You dont know. The situation is perfectly symmetric; picking and switching is the same as just picking the second one at first. So just pick one.
- colomon 13y ago
  Env A has $20 (we have it and know). Env B has $10 or $40 (equal chance). We swap. Now we hold Env B, which has $10 or $40 in it. We know the other envelope (Env A) is $20. Why would we switch again?
  - andrewflnr 13y ago
    The original statement of the problem says you're offered the choice before you open the envelope. I guess I was sloppy about the $20 assumption too. As for the case you do know, I still feel wrong about the conclusion that you should switch, but I can't formalize it.