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The two envelopes problem
Posted: 18 November 2012 10:50 PM   [ Ignore ]   [ # 1591 ]
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Mingy Jongo - 18 November 2012 05:44 PM

Kkwan has literally been repeating the same “your envelope is X and the other is either 2X or 1/2X with 50-50 probability” nonsense for almost a year now, despite having been shown time and time again that mathematics doesn’t work that way when dealing with unknown probability distributions.  Isn’t it time for everyone to move on?

How do you do the puzzle if we open our envelope?

What I would expect is to be able to do an equation that shows that the expected gain is the same as the expected loss, but the only equation I can think of that produces the result is your envelope is X and the other is 2X or 1/2X with 2 in 3 and 1 in 3 probability, which obviously is wrong. 

I think this is what keeps the puzzle going as it appears that opening the envelope makes no difference.

Stephen

[ Edited: 18 November 2012 11:00 PM by StephenLawrence ]
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Posted: 18 November 2012 11:11 PM   [ Ignore ]   [ # 1592 ]
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TimB - 18 November 2012 09:37 PM

No! Mingy.  The thread must go on.  If Kkwan were not caught up in this nightmarish perpetuity, he might be using his superior intellect to take over the world.  That must not happen.

Exactly. You all do not realise the danger we are in. Imagine what could happen with our monetary system! When this master mind gets everybody increasing his savings by putting them into envelopes and switching them endlessly it would completely break down! I must go on creating confusion (my own denialist project. I should contact the tobacco industry and Koch Industries…) to avoid worse happening! Be with me, to avoid this economic catastrophe!

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Posted: 18 November 2012 11:23 PM   [ Ignore ]   [ # 1593 ]
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StephenLawrence - 18 November 2012 10:50 PM

I think this is what keeps the puzzle going as it appears that opening the envelope makes no difference.

No, that is what keeps you interested. For the rest kkwan is either just amusing himself by repeating the silly stuff over and over again, or just too stubborn to confess that he was wrong all the time.

Concerning your interest. As said, I answered to that here.
Your objection here is not true. The calculation still stands when you open your envelope.

When you think about the situation, be sure you have taken everything in account: that there are two envelopes between which you choose first, open the envelope, and then decide if you take the other one or not. This is not the same as: you get an envelope, you open it and see the amount, and then a second envelope is presented to you that contains half or twice the amount that you have now. Don’t you see these are different situations? Don’t you see that you must treat them differently?

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Posted: 18 November 2012 11:34 PM   [ Ignore ]   [ # 1594 ]
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GdB - 16 November 2012 01:47 AM
StephenLawrence - 16 November 2012 12:04 AM
GdB - 15 November 2012 07:30 AM

Then you must treat them as mutually exclusive, but you don’t. See here where I do that. The result is that switching does not help.

rolleyes

But this ceases to work if you open your envelope. And you say opening your envelope makes no difference.

Why? I start with “you open the envelope and you see the amount $X”.

The problem is if we work with say $10 and $20 or $20 and $40 we have an equation that shows expected gain is the same as the expected loss as long as we don’t specify which of these numbers we have.

If we have $40 we’ll lose $20 and if we have $10 we’ll only gain $10. What you are doing is using this to cancel out Kkwan’s equation.

But this fails if we open the envelope and find $20. You no longer have anything to cancel out the apparent expected gain and you say opening the envelope makes no difference.

Because of this your solution is wrong.

Stephen

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Posted: 18 November 2012 11:44 PM   [ Ignore ]   [ # 1595 ]
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GdB - 18 November 2012 11:23 PM
StephenLawrence - 18 November 2012 10:50 PM

I think this is what keeps the puzzle going as it appears that opening the envelope makes no difference.

No, that is what keeps you interested.

P’raps you’re right.

Still I think we’ll get an answer when we understand what’s going on with our envelope open and at the moment we don’t.

Until then your answer is kkwan’s equation is wrong, which I don’t think counts as a solution.

Stephen

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Posted: 19 November 2012 12:36 AM   [ Ignore ]   [ # 1596 ]
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StephenLawrence - 18 November 2012 11:34 PM

But this fails if we open the envelope and find $20. You no longer have anything to cancel out the apparent expected gain and you say opening the envelope makes no difference.

Because of this your solution is wrong.

No. Realise what you say here: you say that your intuitions fail.

But to make the calculation, you must take all the circumstances into account. So if you see the amount, and find $20, you must take into account that you could have had two different pairs (10,20) and (20,40). But then you must do the calculation as I did here. Tell me where it is wrong, and I will react. All the other stuff is your intuitions running astray.

Again think about the two different situations, and how to treat them correctly: that there are two envelopes between which you choose first, open the envelope, and then decide if you take the other one or not. This is not the same as: you get an envelope, you open it and see the amount, and then a second envelope is presented to you that contains half or twice the amount that you have now.

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Posted: 19 November 2012 01:10 AM   [ Ignore ]   [ # 1597 ]
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GdB - 19 November 2012 12:36 AM

No. Realise what you say here: you say that your intuitions fail.

Nope, my intuitons have nothing to do with it.

But to make the calculation, you must take all the circumstances into account. So if you see the amount, and find $20, you must take into account that you could have had two different pairs (10,20) and (20,40). But then you must do the calculation as I did here. Tell me where it is wrong, and I will react. All the other stuff is your intuitions running astray.

I’ve told you exactly what is wrong. You’re using the fact that you might have had $10 or $40 in your envelope to cancel out the expected gain if you have $20 in your envelope with an equivalent expected loss.

You can’t do that because you know you have $20 in your envelope.

The fact is you have no equation that shows the expected gain is the same as the expected loss, the only equation that works is that you have only a 1 in 3 chance that $20 is the smaller amount, so the expected gain is equivalent to the expected loss.

Stephen

[ Edited: 19 November 2012 01:13 AM by StephenLawrence ]
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Posted: 19 November 2012 01:29 AM   [ Ignore ]   [ # 1598 ]
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StephenLawrence - 19 November 2012 01:10 AM
GdB - 19 November 2012 12:36 AM

No. Realise what you say here: you say that your intuitions fail.

Nope, my intuitons have nothing to do with it.

Well, yes. What is that ‘cancel out’ you are using below?? Sit down and calculate! But begin at the beginning: two closed envelopes, and you can pick one. Start there. If you don’t, your results will be wrong, as in the Monty Hall problem.

StephenLawrence - 19 November 2012 01:10 AM

I’ve told you exactly what is wrong. You’re using the fact that you might have had $10 or $40 in your envelope to cancel out the expected gain if you have $20 in your envelope with an equivalent expected loss.

You can’t do that because you know you have $20 in your envelope.

No idea what you are saying here. If I have $20, then the original pair was ($10, $20) or ($20,$40): assuming equal chances the chance of getting $10 was 1/4, the chance of getting $40 is 1/4, and of getting $20 is 1/2. What is wrong with that?

StephenLawrence - 19 November 2012 01:10 AM

The fact is you have no equation that shows the expected gain is the same as the expected loss, the only equation that works is that you have only a 1 in 3 chance that $20 is the smaller amount, so the expected gain is equivalent to the expected loss.

I have no idea what equation you want. If I have two amounts, then on switching the possible gain and loss are the same. That is still so if I know one of the amounts.

Stephen, work out the difference between TEP, and having one known amount and another that is half or twice the amount you have.

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Posted: 19 November 2012 01:49 AM   [ Ignore ]   [ # 1599 ]
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GdB - 19 November 2012 01:29 AM

Well, yes. What is that ‘cancel out’ you are using below?? Sit down and calculate!

I have. If you have $10 you will gain $10 dollars. If you have $20 And it’s the smaller amount you will gain $20. If you have £20 and it is the larger amount you will lose $10. If you have $40 you will lose $20.

The red loss cancels out the black gain.

But begin at the beginning: two closed envelopes, and you can pick one. Start there. If you don’t, your results will be wrong, as in the Monty Hall problem.

Simply no GdB. You’ve made the assumption that there is equal chance of having $10 and $20 or $20 and $40 and that is all there is to it, once you make that assumption we should switch.

No idea what you are saying here. If I have $20, then the original pair was ($10, $20) or ($20,$40): assuming equal chances the chance of getting $10 was 1/4, the chance of getting $40 is 1/4, and of getting $20 is 1/2. What is wrong with that?

That’s correct. It’s also correct that assuming equal chances we should switch. And the reason is you cannot use the red calculations to cancel out the black calculations. That is what you are doing.

 

I have no idea what equation you want. If I have two amounts, then on switching the possible gain and loss are the same. That is still so if I know one of the amounts.

I want an equation that uses the fact we have $20 and the fact we could have $10 or $40 in the other envelope and comes out with the expected gain equivalent to the expected loss. So one example is to make the probability of having $20 and $40 1 in 3 and the probability of having $10 and $20 2 in 3.

Stephen

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Posted: 19 November 2012 01:51 AM   [ Ignore ]   [ # 1600 ]
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Stephen, work out the difference between TEP, and having one known amount and another that is half or twice the amount you have.

Simply no GdB. You’ve made the assumption that there is equal chance of having $10 and $20 or $20 and $40 and that is all there is to it, once you make that assumption we should switch.

Why?

[ Edited: 19 November 2012 01:54 AM by GdB ]
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Posted: 19 November 2012 01:57 AM   [ Ignore ]   [ # 1601 ]
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GdB - 19 November 2012 01:51 AM

Stephen, work out the difference between TEP, and having one known amount and another that is half or twice the amount you have.

GdB,

Fact is your solution is wrong and I’ve said why that is clearly enough for you to know that.

I dunno the right solution, that’s why I’m interested.

What you’ve said is there is no difference. Because, of course if we open our envelope we do have one know amount and another that is half or twice the amount I have.

Stephen

[ Edited: 19 November 2012 02:44 AM by StephenLawrence ]
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Posted: 19 November 2012 02:01 AM   [ Ignore ]   [ # 1602 ]
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StephenLawrence - 19 November 2012 01:57 AM

What you’ve said is there is no difference. Because, of course if we open our envelope we have do have one know amount and another that is half or twice the amount I have.

So you treat TEP the same as the situation where you get one amount, and then you are offered a second envelope with half or twice the amount. Where do you use the knowledge you have that you originally had 2 envelopes to choose from, and that the choice you have is to pick the other one instead? You do not use it.

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Posted: 19 November 2012 02:01 AM   [ Ignore ]   [ # 1603 ]
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GdB - 19 November 2012 01:51 AM

Simply no GdB. You’ve made the assumption that there is equal chance of having $10 and $20 or $20 and $40 and that is all there is to it, once you make that assumption we should switch.

Why?

Because what that means is if I have $20 I have equal chance of the two envelopes containing $20 and $40 or $10 and $20.

Which means by switching I end up with an equal chance of ending up with $10 or $40.

Stephen

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Posted: 19 November 2012 02:05 AM   [ Ignore ]   [ # 1604 ]
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StephenLawrence - 19 November 2012 02:01 AM

Because what that means is if I have $20 I have equal chance of the two envelopes containing $20 and $40 or $10 and $20.

Which means by switching I end up with an equal chance of ending up with $10 or $40.

Then tell me which step in my calculation is wrong.

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Posted: 19 November 2012 02:06 AM   [ Ignore ]   [ # 1605 ]
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GdB - 19 November 2012 02:01 AM
StephenLawrence - 19 November 2012 01:57 AM

What you’ve said is there is no difference. Because, of course if we open our envelope we have do have one know amount and another that is half or twice the amount I have.

So you treat TEP the same as the situation where you get one amount, and then you are offered a second envelope with half or twice the amount.

The fact is, with your envelope open that is the situation.

Where do you use the knowledge you have that you originally had 2 envelopes to choose from, and that the choice you have is to pick the other one instead? You do not use it.

Then find the correct way to apply it.

Stephen

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