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The two envelopes problem
 Posted: 22 November 2012 03:21 AM [ Ignore ]   [ # 1666 ]
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StephenLawrence - 22 November 2012 12:25 AM

What is wrong is you are assuming that we can gain double or lose half with 50/50 probability.

And coming to the conclusion that we shouldn’t switch.

Sorry Stephen, this translates to “because it does not fit my intuitions, it is wrong”.

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 Posted: 22 November 2012 03:23 AM [ Ignore ]   [ # 1667 ]
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TimB - 22 November 2012 12:27 AM

Well if those are the only 2 options, I hope that it is the former, though I don’t wish CFI to be closed down.  Maybe if he gets exhorbitantly wealthy, he will keep the forum going, and particularly this thread, so as to continuously throw it in everyone’s face that he was right all along.

God save us from such a scenario!

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 Posted: 22 November 2012 03:27 AM [ Ignore ]   [ # 1668 ]
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StephenLawrence - 22 November 2012 01:06 AM

On seeing what’s in our envelope we can’t get specific about that situation because we don’t know the probability of having double or half.

We know we had the choice between the two envelopes, and is the information you do not use.
If you think you do, then show me.

At least you should see now that the TEP scenario yields different results, then the scenario where you get one envelope and then the other envelope is filled with twice or half of the amount you have. That should give you the clue.

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 Posted: 22 November 2012 10:05 AM [ Ignore ]   [ # 1669 ]
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GdB - 22 November 2012 03:21 AM
StephenLawrence - 22 November 2012 12:25 AM

What is wrong is you are assuming that we can gain double or lose half with 50/50 probability.

And coming to the conclusion that we shouldn’t switch.

Sorry Stephen, this translates to “because it does not fit my intuitions, it is wrong”.

Well, the solution to the TEP using your example is to say why it isn’t a contradiction.

The reason is because although we can gain double or lose half with a probability of 50/50 in this situation we don’t know it. If we had that information we should switch but we don’t.

Because we don’t have the information about the probability distribution, knowing what we have in our envelope does not help us at all. We cannot use the information.

It’s because we cannot use the information that we should treat the situation as if we don’t know what we picked.

This is now a pretty full explanation.

If that is what you were trying to say I agree.

Stephen

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 Posted: 22 November 2012 10:53 AM [ Ignore ]   [ # 1670 ]
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GdB - 22 November 2012 03:23 AM
TimB - 22 November 2012 12:27 AM

Well if those are the only 2 options, I hope that it is the former, though I don’t wish CFI to be closed down.  Maybe if he gets exhorbitantly wealthy, he will keep the forum going, and particularly this thread, so as to continuously throw it in everyone’s face that he was right all along.

God save us from such a scenario!

On this forum, it might be better to say “Humanity save us from such a scenario!”

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 Posted: 23 November 2012 12:04 AM [ Ignore ]   [ # 1671 ]
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Stephen,

The ground problem is that chance is not just something subjective. You must use the objective view, and that is that you are presented a TEP pair, and then you pick one envelope. It goes without saying that the envelope you did not choose is the one you can switch to. So you must take the pair as it is as you starting point, and not your subjective position. That is an error in the Monty Hall problem, and it is here as well. It is also the problem of kkwan. (You cannot become ‘objective rich’ by taking a ‘subjective position’). He takes the envelope that he has picked first as fix point, and that is wrong. The fix point is that there are two different amounts.

All TEP pairs can presented as (X,2X). So in general the solution is then playing through the two possible situations:

- You pick the envelope with X first. You may even look at the amount, it makes no difference. If you switch you gain X.
- You pick the envelope with 2X first. You may even look at the amount, it makes no difference. If you switch you loose X.

The subject view is that you have a certain amount, X, and assuming that the chance that the pair could have been (X,X/2) or (X,2X) are equal, you have an equal chance of finding X/2 or 2X in the other envelope. But by presenting it as such you have not used the information that you are starting by picking one amount of one pair. The two possibilities present themselves to you as

-  You pick an envelope, you see the amount, X, the other contains twice as much so you gain X
-  You pick an envelope, you see the amount, X, the other contains half of it so you loose X/2

So it is the same error again and again: taking the envelope of your first choice as a constant.

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 Posted: 23 November 2012 12:47 AM [ Ignore ]   [ # 1672 ]
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GdB - 23 November 2012 12:04 AM

The subject view is that you have a certain amount, X, and assuming that the chance that the pair could have been (X,X/2) or (X,2X) are equal, you have an equal chance of finding X/2 or 2X in the other envelope. But by presenting it as such you have not used the information that you are starting by picking one amount of one pair.

Let’s say I open my envelope and find \$20 and there is \$40 in the other envelope.

The objective view is I should switch.

We can’t use that objective view because we don’t have that information.

The next nearest objective view is I have \$20 and there might be \$10 or \$40 in the other envelope.

But we can’t use that because we don’t know the probability distribution.

Stephen

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 Posted: 23 November 2012 01:15 AM [ Ignore ]   [ # 1673 ]
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StephenLawrence - 23 November 2012 12:47 AM

Let’s say I open my envelope and find \$20 and there is \$40 in the other envelope.

The objective view is I should switch.

We can’t use that objective view because we don’t have that information.

The next nearest objective view is I have \$20 and there might be \$10 or \$40 in the other envelope.

No. The next objective view is that you picked your envelope from two envelopes, and that the other contains the remaining amount.

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 Posted: 23 November 2012 01:26 AM [ Ignore ]   [ # 1674 ]
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StephenLawrence - 23 November 2012 12:47 AM

The next nearest objective view is I have \$20 and there might be \$10 or \$40 in the other envelope.

Forgot to say: by doing this you do as if you are in the second scenario. But you know you are not in that scenario.

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 Posted: 23 November 2012 01:29 AM [ Ignore ]   [ # 1675 ]
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GdB - 23 November 2012 01:15 AM

The next objective view is that you picked your envelope from two envelopes, and that the other contains the remaining amount.

Sure and that remaining amount might be \$40 or \$10 which is as objective as we can get.

Really I just think bringing objectivity into it makes no sense but if you are going to the above view is as objective as it gets.

Stephen

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 Posted: 23 November 2012 01:32 AM [ Ignore ]   [ # 1676 ]
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GdB - 23 November 2012 01:26 AM
StephenLawrence - 23 November 2012 12:47 AM

The next nearest objective view is I have \$20 and there might be \$10 or \$40 in the other envelope.

Forgot to say: by doing this you do as if you are in the second scenario. But you know you are not in that scenario.

It’s t’other way around.

I am in the second scenario but don’t know it.

Things like this are why I don’t feel you have a grasp of the solution.

Stephen

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 Posted: 23 November 2012 01:37 AM [ Ignore ]   [ # 1677 ]
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StephenLawrence - 23 November 2012 01:32 AM

It’s t’other way around.

I am in the second scenario but don’t know it.

No, this is BS. You know we are playing TEP: you know you are not in the second scenario. When you pick one envelope, the other one with the other amount remains. You do as if this is not the case and you are in a whole new situation.

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 Posted: 23 November 2012 11:11 AM [ Ignore ]   [ # 1678 ]
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GdB - 23 November 2012 01:37 AM
StephenLawrence - 23 November 2012 01:32 AM

It’s t’other way around.

I am in the second scenario but don’t know it.

No, this is BS. You know we are playing TEP: you know you are not in the second scenario. When you pick one envelope, the other one with the other amount remains. You do as if this is not the case and you are in a whole new situation.

Actually, if you recall, my equation was based on the TEP type history you gave me.

It does fit with the ‘objective’ history and would have been correct if I had known the history.

So definately not BS GdB.

The relevant different between the two scenarios is in one case I know the probability distribution and in the other case I don’t.

This is key to a proper solution to the problem.

Stephen

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 Posted: 23 November 2012 12:43 PM [ Ignore ]   [ # 1679 ]
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Occam. - 21 November 2012 07:24 PM

I’m guessing you’re referring to your post #1641 on the prior page (110).  If you go to the top of the page to the box showing that this is page 111 and click on 110, you should be able to see your earlier post.

Occam

thanks, I did find it.
....

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 Posted: 23 November 2012 12:48 PM [ Ignore ]   [ # 1680 ]
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GdB - 21 November 2012 06:57 AM
Lois - 21 November 2012 06:15 AM

(Someone else in this thread might have mentioned vos Savant and these websites, but I looked at this thread only recently amd didn’t feel up to going through all 110 posts.)

Pages. More than 1600 posts.

Thanks. You’re right, which makes it even more unproductive to attempt to go through them all. Better to just risk making the same point twice.

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