Talk:Two envelopes problem/Archive 1

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The following was copied from Talk:Monty Hall problem Mintguy (T) 09:45, 27 Aug 2004 (UTC)

I was talking to a friend about the Monty Hall problem today and he told me about a similar problem, and I don't think there is a Wikipedia article on it and I'm not sure how to present the solution either, but anyway here is the problem:

You are on a gameshow and the host holds out two envelopes for you to choose from A and B. So you choose an envelope (A) and it's got $2000 in it. The presenter then says that one of the envelopes has twice as much money in it as the other one and offers you the chance to switch. So you think about it this way... "If I switch I will go home with either $4000 or $1000, by not switching I will go home with $2000. There is a 50/50 chance that I will double my money by switching. A normal 50/50 bet results in me either doubling my money or losing it all, whereas here I will only lose half. Therefore this is a better than evens bet so I will make the swap." You are just about to swap envelopes when you think about the problem some more - "Surely this can't be right... ". Mintguy (T) 16:13, 15 Jul 2004 (UTC)

Interesting, but not really similar. In the Monty Hall problem, there's just one possible positive payoff, so the only concern is maximizing your chance of getting it. This problem is more complicated. It is true that if you switch, the expected value of the new envelope is (1000*0.5 + 4000*0.5) = 2,500, so if all you care about is the average amount of money you'll take home, you should switch. However, most people in the real world are risk averse, meaning that they may prefer the sure 2,000. Isomorphic 02:58, 16 Jul 2004 (UTC)
Yeah but you're wrong, you see, there is no advantage to switching. How can there be? . If you didn't know how much money was in the envelope, then you might make the same analysis and switch, but then after switching the same analysis would lead you to switch again.Mintguy (T) 07:41, 16 Jul 2004 (UTC)
After I open the first envelope and choose to switch, but before I open the second, here is my position: The other envelope has $2,000. The one I'm holding has 1,000 or 4,000 with (we are assuming) equal probability. My expectation if I switch back is $500 less than if I hold. I'm holding. Dandrake 18:56, Aug 26, 2004 (UTC)

[Tired of nesting the blocks deeper and deeper, like bad C code.] Let's pause to list assumptions. My choice of an envelope is not correlated with the loading of the envelopes, so that I'm equally likely to have the good or the bad envelope. The host also is statistically unbiased: his telling me the news is not correeated with my initial choice of good or bad envelope. Oh, and he's telling the truth.

This is not to say that the expectation argument is right. This is the paradoxical part: that the argument has no apparent flaw in itself, but it gives the nonsensical result that one should choose and then change, even though no new information has come along to cause a change. The host's information is new, or seems to be, but how would my course of action be different if I had known it all along? Bottom line: the expectation argument leads to a silly result, but I don't believe that its flaw has been shown. Maybe this paradox deserves its own article. Dandrake 19:35, Aug 26, 2004 (UTC)


I think I'm not understanding this problem correctly. The way I read it, you either pick an envelope with X or 2X dollars in it with equal probability. Given the option to switch, this expands into four cases

Picked X Picked 2X
Keep X 2X
Switch 2X X

Doesn't this mean that each of these events occurs with equal probablity, and that it doesn't matter? I understand the expectation argument, but I can't reconcile it with this simple grid. Cvaneg 23:13, 26 Aug 2004 (UTC)

I think that that is the point. The paradox lies in reconciling the argument with the grid or finding the flaw in the argument. No one is denying that the grid gives the correct answer. The trouble is that the expectation argument seems reasonable and yet gives the wrong answer. -- Derek Ross | Talk 16:27, 2004 Oct 5 (UTC)