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The two envelopes problem
 Posted: 16 May 2012 11:12 AM [ Ignore ]   [ # 916 ]
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StephenLawrence - 16 May 2012 11:05 AM

Trouble is you’ve changed it to known amounts GdB. So it’s totally understandable for us to just think you were wrong.

OK.

So you agree that there are \$10 and \$20 in the envelopes, when I put \$10 and \$20 in it?

What is in the envelopes when you see that I put these amounts into the envelopes?

What is in the envelopes when you do not see that I put these amounts into the envelopes?

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 Posted: 16 May 2012 11:14 AM [ Ignore ]   [ # 917 ]
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GdB - 16 May 2012 10:34 AM

If there are \$10 and \$20 in the envelopes, and you pick the one with \$20 first, is there then \$40 in one of the envelopes?

This appears to be no more relevent than asking if you have the one with \$20 do you have the one with \$10. Of course you don’t have the one with \$10 but you might, which is what matters.

Equally of course there isn’t \$40 in one of the envelopes but there might be.

This really seems to be going nowhere.

Stephen

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 Posted: 16 May 2012 11:38 AM [ Ignore ]   [ # 918 ]
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GdB - 16 May 2012 11:12 AM

So you agree that there are \$10 and \$20 in the envelopes, when I put \$10 and \$20 in it?

What is in the envelopes when you see that I put these amounts into the envelopes?

\$10 and \$20

What is in the envelopes when you do not see that I put these amounts into the envelopes?

\$10 and \$20

Stephen

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 Posted: 16 May 2012 11:43 AM [ Ignore ]   [ # 919 ]
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StephenLawrence - 16 May 2012 11:14 AM

This really seems to be going nowhere.

Gee, you think?

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 Posted: 16 May 2012 01:06 PM [ Ignore ]   [ # 920 ]
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StephenLawrence - 16 May 2012 11:38 AM
GdB - 16 May 2012 11:12 AM

So you agree that there are \$10 and \$20 in the envelopes, when I put \$10 and \$20 in it?

What is in the envelopes when you see that I put these amounts into the envelopes?

\$10 and \$20

What is in the envelopes when you do not see that I put these amounts into the envelopes?

\$10 and \$20

Right. Maybe we get somewhere. (Get lost, FreeInKy!  )

So again: if I put \$10 and \$20 in the envelopes, what is the chance that there is \$40 in one of the envelopes?
And what there is \$5 in it?

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 Posted: 16 May 2012 01:07 PM [ Ignore ]   [ # 921 ]
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FreeInKy - 16 May 2012 11:43 AM
StephenLawrence - 16 May 2012 11:14 AM

This really seems to be going nowhere.

Gee, you think?

You’ll never know, FreeInKy, you’ll never know…

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 Posted: 16 May 2012 09:55 PM [ Ignore ]   [ # 922 ]
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GdB - 16 May 2012 10:34 AM

I have two amount in two envelopes, X and 2X. How much is in one envelope, and how much in the other?
Is X a constant?
Do 2 amounts, X and 2X, fit to the conditions of TEP? (One envelope contains twice as much as the other. )

In the TEP, the amounts in the two envelopes are unknown.

As such, you can only imagine (X, 2X) are in the envelopes.

X is a constant if a value is assigned to it. If not, it is a variable as a symbol that represents an unknown quantity in an algebraic expression.

Yes, but so does (X, 1/2X) and it is also possible but you have omitted it.

If there are \$10 and \$20 in the envelopes, and you pick the one with \$10 first, is there then \$5 in one of the envelopes?

In the TEP, you don’t know what are in the envelopes, so you cannot assume (\$10, \$20) are in the envelopes. You can only imagine (\$10, \$20) are in the envelopes, but it is also possible that (\$10, \$5) are in the envelopes if you selected an envelope and it contains \$10.

If there are \$10 and \$20 in the envelopes, and you pick the one with \$20 first, is there then \$40 in one of the envelopes?

Firstly, in the TEP, you don’t know what are in the envelopes and thus you cannot assume (\$10, \$20) or any amounts, are in the envelopes.

As such, you can only imagine (\$10, \$20) but if you do so, it is also possible (\$10, \$5) are in the envelopes, if the amount in the selected envelope is \$10.

OTOH, if the amount in the selected envelope is \$20, it is either (\$20, \$40) or (\$20, \$10) in the envelopes.

In either case, (\$10, \$20) is not exhaustive of all the possibilities and thus it is one-sided, incomplete and misleading.

Let me ask a little different: what is the expected value in the TEP-like situation when you switch? Don’t forget: the ‘play master’ gives you the envelope with the ‘middle amount’, he knows the amounts, so he can prepare the envelopes with Z/2, Z (the amount that you get), and 2Z.

Your TEP-like situation with three amounts in three envelopes is not the TEP per se.

As such, it has no relevance to the TEP and it is the TEP we are discussing.

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 Posted: 16 May 2012 11:06 PM [ Ignore ]   [ # 923 ]
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GdB - 16 May 2012 01:06 PM

So again: if I put \$10 and \$20 in the envelopes, what is the chance that there is \$40 in one of the envelopes?
And what there is \$5 in it?

Assuming the numbers are selected at random the same chance as there is \$10 in one of the envelopes and the same chance as there is \$20 in one of the envelopes.

It’s no difference than asking (edit) if I toss a coin and it lands on tails what is the chance of it being a head. The answer is the same as the chance of it being a tail.

Stephen

[ Edited: 16 May 2012 11:18 PM by StephenLawrence ]
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 Posted: 16 May 2012 11:15 PM [ Ignore ]   [ # 924 ]
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FreeInKy - 16 May 2012 11:43 AM
StephenLawrence - 16 May 2012 11:14 AM

This really seems to be going nowhere.

Gee, you think?

Well there should be a solution, it’s just GdB is seemingly switching the situation from unknown to known to try to make his solution work, which is against the rules.

Perhaps he could rescue the situation by saying we can treat the puzzle as if the amounts are known because….............

Something along those lines could convince me but it’s hard to imagine what that something might be.

Stephen

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 Posted: 17 May 2012 12:05 AM [ Ignore ]   [ # 925 ]
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StephenLawrence - 16 May 2012 11:15 PM

Well there should be a solution, it’s just GdB is seemingly switching the situation from unknown to known to try to make his solution work, which is against the rules.

Perhaps he could rescue the situation by saying we can treat the puzzle as if the amounts are known because….............

Something along those lines could convince me but it’s hard to imagine what that something might be.

I don’t have to ‘rescue’ the situation. I already gave my solution (yes, the one with the X’s): the answer is correct, and I say to you, the derivation is too. You asked about a hundred postings ago how I can know that my solution is correct, independent of it ‘accidentally’ giving the correct answer, where kkwan’s solution seems to be correct, but gives the wrong answer. This is answered already many times, not just by me: the unknowns used in kkwan’s formula do not have the same value, and therefore cannot be used as constants in his formula.

The point I am driving at in my recent childish questions, is to give you the feeling that when there are two amounts in the envelopes, that might be unknown by ‘the player’, but the amount that are in the envelopes are in it. They are closed, and what is in one envelope or the other does not change if I pick one envelope first, or then the other.

So IF, by example, there are \$10 and \$20 in the envelopes, independent of the knowledge of the player about it, there will never be \$5 or \$40 in one of them. We do not have to think about it.

The player knows that, even if he does not know the amounts, there are only 2 amounts in the envelopes, and therefore can be exhaustively described by X and 2X. If you keep hesitating about this, imagine two values that fit TEP, and see that there are only two amounts. (Come on, do it, take two envelopes and play the game with your spouse/son/daughter/friend!) So the only 2 possibilities are that you pick the smallest amount first, or the largest, i.e. X or 2X. Just think about it: there are two amounts, the smaller and the bigger. But kkwan states there is a third amount that I should reckon with. Does that make sense to you?

To show that, I gave another, TEP-like, situation that fits to kkwan’s derivation. (Read it a few postings back.) The question ‘what is the expected value?’ can be calculated exactly with kkwan’s formula. But it is not TEP anymore. As kkwan is not able to show why his formula does not apply to my TEP-like situation, and by saying this is not TEP, he confirms my point: his formula is not valid in real-TEP.

My only point here is that I am explaining this to you both. You are the only ones in the school class that do not see the point. Your colleagues are home already for a long time, but the teacher tries to give the correct insight to the two last pupils who resist to what is already clear for all the others. One pupil thinks that everything must be more complicated than the teacher says (he has trouble to understand the solution, that it can be that simple), the other one will never give in, because he put all his personal credibility in his wrong derivation.

Listen, I have some mathematical background, and you haven’t both. There was another mathematician on this thread who better than I explained and derived the correct solution, and you both did not recognise it, be it by lack of insight, or by just wanting to be awkward, by trolling.

PS This topic was already in the wrong forum from the beginning. As it is a mathematical problem, and mathematics is a science, it would belong in ‘Science and technology’.

[ Edited: 17 May 2012 07:09 AM by GdB ]
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 Posted: 17 May 2012 02:19 AM [ Ignore ]   [ # 926 ]
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GdB - 17 May 2012 12:05 AM

The point I am driving at in my recent childish questions, is to give you the feeling that when there are two amounts in the envelopes, that might be unknown by ‘the player’, but the amount that are in the envelopes are in it. They are closed, and what is in one envelope or the other does not change if I pick one envelope first, or then the other.

Which hasn’t helped at all, of course the amounts don’t change. Look in your envelope and it contains \$10, still the amount in the other envelope doesn’t change but now you say it could be \$5 or \$20. If it could be after you look it could be before you look too.

So IF, by example, there are \$10 and \$20 in the envelopes, independent of the knowledge of the player about it, there will never be \$5 or \$40 in one of them. We do not have to think about it.

What we need to think about is that if there is \$10 and \$20 there could be \$5 and \$10 and there could be \$20 and \$40.

And you agree if we look and find \$10. (not that there could be \$20 and \$40 of course.)

My only point here is that I am explaining this to you both.

Stephen

[ Edited: 17 May 2012 02:32 AM by StephenLawrence ]
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 Posted: 17 May 2012 02:23 AM [ Ignore ]   [ # 927 ]
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StephenLawrence - 17 May 2012 02:19 AM

What we need to think about is that if there is \$10 and \$20 there could be \$5 and \$10 and there could be \$20 and \$40.

Come on. You are saying, if there are \$10 and \$20 in the envelopes there could be \$5 or \$40 in them?

So I put \$10 and \$20 in the envelopes, without you seeing it, and then you say it is possible that there are \$5 or \$40 in one of the envelopes?

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 Posted: 17 May 2012 02:29 AM [ Ignore ]   [ # 928 ]
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GdB - 17 May 2012 02:23 AM
StephenLawrence - 17 May 2012 02:19 AM

What we need to think about is that if there is \$10 and \$20 there could be \$5 and \$10 and there could be \$20 and \$40.

Come on. You are saying, if there are \$10 and \$20 in the envelopes there could be \$5 and \$40 into it?

So I put \$10 and \$20 in the envelopes, without you seeing it, and then you say it is possible that there are \$5 or \$40 in one of the envelopes?

Of course GdB.

Come On.  You are saying is = Could.

So if you have the smaller amount in your envelope, you could not have the larger amount?

Stephen

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 Posted: 17 May 2012 03:22 AM [ Ignore ]   [ # 929 ]
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StephenLawrence - 17 May 2012 02:29 AM
GdB - 17 May 2012 02:23 AM
StephenLawrence - 17 May 2012 02:19 AM

What we need to think about is that if there is \$10 and \$20 there could be \$5 and \$10 and there could be \$20 and \$40.

Come on. You are saying, if there are \$10 and \$20 in the envelopes there could be \$5 and \$40 into it?

So I put \$10 and \$20 in the envelopes, without you seeing it, and then you say it is possible that there are \$5 or \$40 in one of the envelopes?

Of course GdB.

Come On.  You are saying is = Could.

So if you have the smaller amount in your envelope, you could not have the larger amount?

Stephen

StephenLawrence - 16 May 2012 11:38 AM
GdB - 16 May 2012 11:12 AM

So you agree that there are \$10 and \$20 in the envelopes, when I put \$10 and \$20 in it?

What is in the envelopes when you see that I put these amounts into the envelopes?

\$10 and \$20

What is in the envelopes when you do not see that I put these amounts into the envelopes?

\$10 and \$20

Stephen

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GdB

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 Posted: 17 May 2012 04:01 PM [ Ignore ]   [ # 930 ]
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GdB - 17 May 2012 03:22 AM
StephenLawrence - 17 May 2012 02:29 AM
GdB - 17 May 2012 02:23 AM
StephenLawrence - 17 May 2012 02:19 AM

What we need to think about is that if there is \$10 and \$20 there could be \$5 and \$10 and there could be \$20 and \$40.

Come on. You are saying, if there are \$10 and \$20 in the envelopes there could be \$5 and \$40 into it?

So I put \$10 and \$20 in the envelopes, without you seeing it, and then you say it is possible that there are \$5 or \$40 in one of the envelopes?

Of course GdB.

Come On.  You are saying is = Could.

So if you have the smaller amount in your envelope, you could not have the larger amount?

Stephen

StephenLawrence - 16 May 2012 11:38 AM
GdB - 16 May 2012 11:12 AM

So you agree that there are \$10 and \$20 in the envelopes, when I put \$10 and \$20 in it?

What is in the envelopes when you see that I put these amounts into the envelopes?

\$10 and \$20

What is in the envelopes when you do not see that I put these amounts into the envelopes?

\$10 and \$20

Stephen

Huh?

This is very strange.

There are two concepts we are interested in.

1) What is in the envelopes.

2) What could be in the envelopes

You keep appearing to confuse the two so that Is=Could which of course can’t be right because we end up with the probability of absolutely anything being 0 or 1.

Stephen

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