The envelope problem, and a possible solution

[BBB]

[ QUOTE ]
Hi, BBB. You said: "All we are told in the original problem is that one of the envelopes contains twice the other. It tells us nothing about the chances that the other envelope contains twice as much, and to simply assume that these chances are 50% is not correct."

Do you then imply that we choose an envelope in a non random way? Or do you think that those probabilities do not depend on the probability with which you choose the initial envelope? I would think that I have 50% chance op picking up N or 2N.

[/ QUOTE ]

NarobisDad,

You do have a 50% chance of initially picking up on N or 2N. And clearly, once you choose an envelope, there is a 50% chance that the other envelope contains N more than the one you chose, and 50% that it contains N less. If you do not gain or infer any usable information upon viewing the contents of your envelope, then there is still a 50% chance that the other envelope contains N more than yours, and a 50% chance that it contains N less. If you see $100, you now know that N is either $50 or $100. So argument 1 concludes: Since there's a 50% chance that the other envelope contains N more than mine (which is true whether N is $100 or $50), then there's a 50% chance that N contains $200, and a 50% chance that it contains $50. The first part of the preceding statement is correct, BUT THE CONCLUSION IS NOT VALID. The conclusion is only correct if N is equally likely to be $100 or $50 (which may or may not be the case).

If we have some basis of information on which to determine the probability that N is $100 versus the probability that it is $50, for example using what we know in general about how people such as our benefactor might be willing to put in envelopes and give away, it turns out that we should clearly switch if we determine that the probability that N is $100 is more than half the probability that N is $50, and we should clearly not switch if N is more that twice as likely to be $50 as it is to be $100. But to simply guess that N is just as likely to be $100 as it is to be $50 and going from there is totally baseless and meaningless. That would be like if I told you I had a coin in my pocket that was not necessarily fair and I asked you what were the chances that it would come up heads if I flipped it. Unless you had some kind of information on which to determine what the coin might be like, it would be totally meaningless for you to gess 1/2 just because that would be the answer if the coin were fair.