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The two envelopes problem
 Posted: 19 November 2012 02:10 AM [ Ignore ]   [ # 1606 ]
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GdB - 19 November 2012 02:05 AM
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.

I’ve told you GdB.  Hidden in your calculation is the assumption that if we have \$20 in our envelope the expected gain is as Kkwan says. You try to use the fact that we could have had \$10 or \$40 in our envelope to cancel this out.

The fact is if we have equal chance of having either pair we should switch given we know we have \$20 in our envelope.

At this point you are in denial.

Stephen

[ Edited: 19 November 2012 02:41 AM by StephenLawrence ]
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 Posted: 19 November 2012 02:51 AM [ Ignore ]   [ # 1607 ]
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StephenLawrence - 19 November 2012 02:06 AM

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

It’s not. It is like saying that the chances in Monty Hall are fifty-fifty.
As I said, your intuitions are dead wrong.

StephenLawrence - 19 November 2012 02:06 AM

Then find the correct way to apply it.

I did.

StephenLawrence - 19 November 2012 02:10 AM

I’ve told you GdB.  Hidden in your calculation is the assumption that if we have \$20 in our envelope the expected gain is as Kkwan says. You try to use the fact that we could have had \$10 or \$40 in our envelope to cancel this out.

Which TEP pair did I have when I have \$20 in my envelope? And yes, the consequence is that I could have had \$10 or \$40, I am doing a chance calculation.

StephenLawrence - 19 November 2012 02:06 AM

The fact is if we have equal chance of having either pair we should switch given we know we have \$20 in our envelope.

At this point you are in denial.

Of course. I deny wrong conclusions based on faulty arguments!

If you look at my calculation, you will see that I gave equal chances to \$10 and \$40. (i.e. to X/2 and 2X).

[ Edited: 19 November 2012 03:15 AM by GdB ]
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 Posted: 19 November 2012 03:23 AM [ Ignore ]   [ # 1608 ]
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GdB - 19 November 2012 02:51 AM
StephenLawrence - 19 November 2012 02:06 AM

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

It’s not. It is like saying that the chances in Monty Hall are fifty-fifty.

No it is not. The chance is 50/50 assuming the chance of having \$10 and \$20 or \$20 and \$40 is 50/50, which is the assumption we are working on for the mo.

.

StephenLawrence - 19 November 2012 02:10 AM

I’ve told you GdB.  Hidden in your calculation is the assumption that if we have \$20 in our envelope the expected gain is as Kkwan says. You try to use the fact that we could have had \$10 or \$40 in our envelope to cancel this out.

Which TEP pair did I have when I have \$20 in my envelope?

\$10 and \$20 or \$20 and \$40 with equal probability is the assumption.

And yes, the consequence is that I could have had \$10 or \$40, I am doing a chance calculation.

Which is fine if we don’t know we have \$20. Once we do know the fact we could have had \$10 or \$40 cannot cancel out the expected gain. Gaining the information that we don’t have those amounts gains us the information that we should switch assuming there is an equal probability of having \$10 and \$20 or \$40 and \$20.

If you look at my calculation, you will see that I gave equal chances to \$10 and \$40. (i.e. to X/2 and 2X). But that does not mean I have equal chancde of gaining \$20 or loosing \$10!

You gave equal chances to having \$10 and \$20 or \$40 and \$20 which does mean you have equal chance of gaining \$20 or losing \$10 assuming you know you have \$20 in your envelope.

Stephen

[ Edited: 19 November 2012 03:26 AM by StephenLawrence ]
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 Posted: 19 November 2012 03:32 AM [ Ignore ]   [ # 1609 ]
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Then I have only simple the solution for you.

With any two amounts, you know that on switching your potential loss and gain are the same. Opening the envelope will not change that. Of your two possible guesses one is just wrong. If you would do repeated TEP games, sometimes you will guess right, and sometimes you will guess wrong. And sometimes your wrong guess will be X, sometimes X/2, and sometimes your correct guess will be X and sometimes it will be X/2. Run over several TEP games, the difference will average out.

[ Edited: 19 November 2012 03:38 AM by GdB ]
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 Posted: 19 November 2012 03:41 AM [ Ignore ]   [ # 1610 ]
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GdB - 19 November 2012 03:32 AM

Then I have only simple the solution for you.

With any two amounts, you know that on switching your potential loss and gain are the same. Opening the envelope will not change that. Of your two possible guesses one is just wrong. If you would do repeated TEP games, sometimes you will guess right, and sometimes you will guess wrong. And sometimes your wrong guess will be 2X, sometimes X, and sometimes your correct guess will be 2X and sometimes it will be X. Run over several TEP games, the difference will average out.

The point of this is to find something out GdB. We know there is no point in switching already.

Try answering this. Say I have \$20 in my envelope, what is the probability of having \$40 in the other envelope and what is the probability of having \$10 in the other envelope?

Stephen

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 Posted: 19 November 2012 03:49 AM [ Ignore ]   [ # 1611 ]
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StephenLawrence - 19 November 2012 03:41 AM

Try answering this. Say I have \$20 in my envelope, what is the probability of having \$40 in the other envelope and what is the probability of having \$10 in the other envelope?

First give all the information I need.

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 Posted: 19 November 2012 04:11 AM [ Ignore ]   [ # 1612 ]
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GdB - 19 November 2012 03:49 AM
StephenLawrence - 19 November 2012 03:41 AM

Try answering this. Say I have \$20 in my envelope, what is the probability of having \$40 in the other envelope and what is the probability of having \$10 in the other envelope?

First give all the information I need.

1) There is either \$10 or \$40 in the other envelope no other amounts are possible given you have \$20 and the rules of the game.

2) You know that although you can gain \$20 or lose \$10 your expected gain is 0.

Stephen

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 Posted: 19 November 2012 04:22 AM [ Ignore ]   [ # 1613 ]
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StephenLawrence - 19 November 2012 04:11 AM

1) There is either \$10 or \$40 in the other envelope no other amounts are possible given you have \$20 and the rules of the game.

2) You know that although you can gain \$20 or lose \$10 your expected gain is 0.

Sorry, this is just too confusing. Do I know that I originally had two envelopes to choose from or not?

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 Posted: 19 November 2012 05:03 AM [ Ignore ]   [ # 1614 ]
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GdB - 19 November 2012 04:22 AM
StephenLawrence - 19 November 2012 04:11 AM

1) There is either \$10 or \$40 in the other envelope no other amounts are possible given you have \$20 and the rules of the game.

2) You know that although you can gain \$20 or lose \$10 your expected gain is 0.

Sorry, this is just too confusing. Do I know that I originally had two envelopes to choose from or not?

You have all the information you need.

What is confusing?

If you switch you will either exchange your envelope with \$20 in it for \$40 or \$10

Your expected gain is 0.

That’s it.

Stephen

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 Posted: 19 November 2012 06:51 AM [ Ignore ]   [ # 1615 ]
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I have no idea what you are pointing at. Should I somehow reverse engineer TEP?

If you switch you will either exchange your envelope with \$20 in it for \$40 or \$10

Your expected gain is 0.

is not a description of TEP. You cannot compare the expected value of the amount you have if you don’t know how you got it. So the ‘expected gain’ makes no sense, as you cannot compare the expected value of the value you have, and the value you would get when you switch. So you must start with the chance of getting \$20, which also gives you values for the chances of the other possibilities, \$10 and \$40, and these logically flow into the calculation of the expected two values, the one you have now, and the one when you switch. There is nothing wrong with my calculation.

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 Posted: 19 November 2012 08:12 AM [ Ignore ]   [ # 1616 ]
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GdB - 19 November 2012 06:51 AM

I have no idea what you are pointing at. Should I somehow reverse engineer TEP?

Yeah why not? You say we know that switching makes no difference on opening our envelope. We know that if we switch we might gain \$20 or lose \$10, so all that is missing is the probability of switching and getting \$40 and the probability of switching and getting \$10. If it were 50/50 we should switch, so we know it must be less than that.

is not a description of TEP. You cannot compare the expected value of the amount you have if you don’t know how you got it.

Why do I need to know how I got it. I have \$20 If I switch I might get \$40 and I might get \$10. Now all I need is the probability of that to decide whether to switch or not.

So the ‘expected gain’ makes no sense, as you cannot compare the expected value of the value you have, and the value you would get when you switch.

What doesn’t make sense is to talk about the expected value of the amount I have. I have \$20 and that’s it.

So you must start with the chance of getting \$20, which also gives you values for the chances of the other possibilities, \$10 and \$40, and these logically flow into the calculation of the expected two values, the one you have now, and the one when you switch. There is nothing wrong with my calculation.

Sorry your calculation is wrong. If I assume that there is 50/50 chance of the two envelopes containing either \$20 and \$40 or \$10 and \$20 I should switch, as I know I can gain \$20 or lose \$10 with a probability of 50/50 as Kkwan says.

Stephen

[ Edited: 19 November 2012 08:20 AM by StephenLawrence ]
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 Posted: 19 November 2012 08:46 AM [ Ignore ]   [ # 1617 ]
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StephenLawrence - 19 November 2012 08:12 AM
GdB - 19 November 2012 06:51 AM

I have no idea what you are pointing at. Should I somehow reverse engineer TEP?

Yeah why not? You say we know that switching makes no difference on opening our envelope. We know that if we switch we might gain \$20 or lose \$10, so all that is missing is the probability of switching and getting \$40 and the probability of switching and getting \$10. If it were 50/50 we should switch, so we know it must be less than that.

You want something impossible: in your idea the situation:
- ‘here is an envelope, look what is in it, and then if you want you can pick the other envelope, which contains half or twice the amount in it’
the same as
- ‘here you have two envelopes one contains twice the amount of the other, you may pick one, then look, and then decide to change’.

Do you think that you have the same chance for gaining in the two different games?

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 Posted: 19 November 2012 09:05 AM [ Ignore ]   [ # 1618 ]
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GdB - 19 November 2012 08:46 AM
StephenLawrence - 19 November 2012 08:12 AM
GdB - 19 November 2012 06:51 AM

I have no idea what you are pointing at. Should I somehow reverse engineer TEP?

Yeah why not? You say we know that switching makes no difference on opening our envelope. We know that if we switch we might gain \$20 or lose \$10, so all that is missing is the probability of switching and getting \$40 and the probability of switching and getting \$10. If it were 50/50 we should switch, so we know it must be less than that.

You want something impossible: in your idea the situation:
- ‘here is an envelope, look what is in it, and then if you want you can pick the other envelope, which contains half or twice the amount in it’
the same as
- ‘here you have two envelopes one contains twice the amount of the other, you may pick one, then look, and then decide to change’.

Do you think that you have the same chance for gaining in the two different games?

Yep.

Stephen

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 Posted: 19 November 2012 09:27 AM [ Ignore ]   [ # 1619 ]
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StephenLawrence - 19 November 2012 09:05 AM
GdB - 19 November 2012 08:46 AM

- ‘here is an envelope, look what is in it, and then if you want you can pick the other envelope, which contains half or twice the amount in it’
the same as
- ‘here you have two envelopes one contains twice the amount of the other, you may pick one, then look, and then decide to change’.

Do you think that you have the same chance for gaining in the two different games?

Yep.

Well, that’s dead wrong.

Situation 1: I, the game master, come and bring you two envelopes and play TEP with you. You take one envelope and it contains \$20. What could the other contain? And what is your chance to gain when you switch?
Situation 2: I, the game master, come and bring you two envelopes and you may choose one. You take one envelope and it contains \$20. I ask you how many it contains, and then put the one of \$10 or \$40 in the other envelope. So what could the envelope contain? And what is your chance to gain when you switch?

Do you still think that you have the same chance for gaining in the two different games?

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 Posted: 19 November 2012 09:40 AM [ Ignore ]   [ # 1620 ]
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GdB - 19 November 2012 09:27 AM

Situation 1: I, the game master, come and bring you two envelopes and play TEP with you. You take one envelope and it contains \$20. What could the other contain?

\$10 or \$40

And what is your chance to gain when you switch?

Dunno because I don’t know what the probability of there being \$10 is and the probability of there being \$40 in the other envelope is. If I had that information I could work it out.

Situation 2: I, the game master, come and bring you two envelopes and you may choose one. You take one envelope and it contains \$20. I ask you how many it contains, and then put the one of \$10 or \$40 in the other envelope. So what could the envelope contain?

\$10 or \$40

And what is your chance to gain when you switch?

Dunno because I don’t know what the probability of there being \$10 is and the probability of there being \$40 in the other envelope is. If I had that information I could work it out.

Do you still think that you have the same chance for gaining in the two different games?

Assuming the same probability of there being \$10 in the other envelope and the same probability of there being \$40 in the other envelope, yes.

Stephen

[ Edited: 19 November 2012 09:44 AM by StephenLawrence ]
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