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The two envelopes problem
Posted: 21 November 2012 10:41 AM   [ Ignore ]   [ # 1651 ]
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StephenLawrence - 21 November 2012 09:34 AM

As you know if I switch when I see $40 I will lose every time in your set up, so according to your calculations I should not switch, though I don’t know that.

But you would switch according to your strategy! 40 means it could have been (20,40) or (40,80). So that means if I repeat my game again and again with (10,20) and (20,40), you would always switch. But the netto result of that is 0.
I think I give up on you. I propose you write a small compute program that simulates this for you, and you will see that you netto gain when switching is 0, as is of course when not switching. Maybe you then feel where your intuition lead you astray.

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Posted: 21 November 2012 10:55 AM   [ Ignore ]   [ # 1652 ]
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GdB - 21 November 2012 10:41 AM
StephenLawrence - 21 November 2012 09:34 AM

As you know if I switch when I see $40 I will lose every time in your set up, so according to your calculations I should not switch, though I don’t know that.

But you would switch according to your strategy! 40 means it could have been (20,40) or (40,80). So that means if I repeat my game again and again with (10,20) and (20,40), you would always switch. But the netto result of that is 0.

Correct, so I would lose every time. That’s the point, it isn’t true that switching makes no difference.

Your calculation is wrong because it says switching makes no difference.

I think I give up on you. I propose you write a small compute program that simulates this for you, and you will see that you netto gain when switching is 0, as is of course when not switching. Maybe you then feel where your intuition lead you astray.

GdB, we both know there is something wrong. Solving the puzzle is to articulate what it is.

Your calculation is wrong because you cannot use it to show switching makes no difference.

There is no calculation you can do to show switching makes no difference if we know what we have in our envelope, except to assign 1/3 probability to the other envelope having double and 2/3 probability to it having half.

Stephen

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Posted: 21 November 2012 11:35 AM   [ Ignore ]   [ # 1653 ]
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StephenLawrence - 21 November 2012 10:55 AM

There is no calculation you can do to show switching makes no difference if we know what we have in our envelope, except to assign 1/3 probability to the other envelope having double and 2/3 probability to it having half.

Sounds nearly as in the Monty Hall problem, isn’t it? But it is wrong.

I give up on you. Clearer as here I cannot explain it, and if you don’t get it… Well, then you don’t. But no moment you pointed at an error in it. Play through the hole process, and find your error. But make it somehow ‘static’ will not work out. Your intuitions are wrong.

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Posted: 21 November 2012 11:45 AM   [ Ignore ]   [ # 1654 ]
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GdB - 21 November 2012 11:35 AM

Well, then you don’t. But no moment you pointed at an error in it.

Of course I have pointed out the error.

In the scenario you gave I should switch if I have $20, though I don’t know that.

The fact is if we played this scenario over and over I would gain by switching when I have $20 compared to a player who doesn’t switch.

You calculations don’t fit with that in any way. They just ignore the fact the envelope is open.

Stephen

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Posted: 21 November 2012 11:48 AM   [ Ignore ]   [ # 1655 ]
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GdB - 21 November 2012 11:35 AM
StephenLawrence - 21 November 2012 10:55 AM

There is no calculation you can do to show switching makes no difference if we know what we have in our envelope, except to assign 1/3 probability to the other envelope having double and 2/3 probability to it having half.

Sounds nearly as in the Monty Hall problem, isn’t it? But it is wrong.

It gives us the answer that switching makes no difference and it takes the number we have in our envelope into account, so assigning those probabilities makes sense from that point of view.

It’s weird that assigning those probabilities would give the right answer, and that’s a way of putting the puzzle, why do they?

Stephen

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Posted: 21 November 2012 12:04 PM   [ Ignore ]   [ # 1656 ]
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Thoughts

Looking does make a difference but we don’t know what the difference is.

As we don’t know what the difference is we need to fall back on a general strategy rather than one specific for the amount we have.

We actually should treat the situation as if we can double or half our money with a 1/3 and 2/3 probability.

What pulls the probability down from the 50/50 we would expect is that over many goes with different amounts, the times we lose half will be half of bigger amounts, on average, than the times we double our money.

Stephen

[ Edited: 21 November 2012 04:16 PM by StephenLawrence ]
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Posted: 21 November 2012 04:13 PM   [ Ignore ]   [ # 1657 ]
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Not to elicit panic, but has anyone noticed that Kkwan has not contributed to this thread in the last few days? ... What is he up to?

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Posted: 21 November 2012 05:17 PM   [ Ignore ]   [ # 1658 ]
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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.

This is odd. I don’t see my post, but there is this response to it, tough the quote is incomplete. Anyone know how this happens?

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Posted: 21 November 2012 07:24 PM   [ Ignore ]   [ # 1659 ]
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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

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Posted: 21 November 2012 11:28 PM   [ Ignore ]   [ # 1660 ]
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TimB - 21 November 2012 04:13 PM

Not to elicit panic, but has anyone noticed that Kkwan has not contributed to this thread in the last few days? ... What is he up to?

He will be all right. He only contributes so now and then, repeating the same stuff over and over again. The best possible case of course would be that he is busy switching envelopes to get rich. Maybe he will smash us all by buying the provider of this forum and close it down…
Or he is sitting at some corner of a street, unwashed, unshaved, left by his wife and family, switching two envelopes again and again, murmuring “it should work, it should work, my argument is airtight, I will be rich, I will be rich…”.

Lois - 21 November 2012 05:17 PM

This is odd. I don’t see my post, but there is this response to it, tough the quote is incomplete. Anyone know how this happens?

That it is incomplete is of course my doing. I just wanted to correct your statement that this thread is rather short… 110 Posts, that’s nothing…
I glanced especially through your Wikipedia link, and it is most interesting to see that so many people do not get loose from their wrong intuitions.

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Posted: 21 November 2012 11:41 PM   [ Ignore ]   [ # 1661 ]
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GdB,

Here is a way of looking at this which is essentially the same as yours.

There are two envelopes X and X2, I pick one and am then asked if I’d like to switch. I reason switching makes no difference.

Then I open my envelope and have $20 dollars.

I then reason as follows:

A

If $20 is the smaller amount I will gain $20 with 50/50 probability

If $20 is the larger amount I will lose $10 with 50/50 probability

So switch .

B

But if $20 is the smaller amount and I had picked $40 I would have lose $20 with 50/50 probability

And if $20 is the larger amount and I had picked $10 I would have gained $10 with 50/50 probability.

B cancels out A so don’t switch.

The fact is it doesn’t matter if B cancels out A because I know I have $20 and if it makes sense to apply the 50/50 probability I know I should switch in the situation I am in.

I think it’s entirely obvious that you can’t do this. All you are doing is treating the situation as if you don’t know what is in your envelope because conveniently you get the answer that switching makes no difference.

The problem has to be with assuming 50/50 probability.

You can’t assume it and conclude don’t switch, which is what you are doing.

Stephen

[ Edited: 21 November 2012 11:44 PM by StephenLawrence ]
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Posted: 22 November 2012 12:25 AM   [ Ignore ]   [ # 1662 ]
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GdB - 21 November 2012 08:39 AM

Stephen, just look at the ‘calculations’ above, and say what is wrong.

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.

Which is a contradictory result.

It’s that simple.

I’ll move on from making that point now.

Stephen

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Posted: 22 November 2012 12:27 AM   [ Ignore ]   [ # 1663 ]
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GdB - 21 November 2012 11:28 PM
TimB - 21 November 2012 04:13 PM

Not to elicit panic, but has anyone noticed that Kkwan has not contributed to this thread in the last few days? ... What is he up to?

He will be all right. He only contributes so now and then, repeating the same stuff over and over again. The best possible case of course would be that he is busy switching envelopes to get rich. Maybe he will smash us all by buying the provider of this forum and close it down…
Or he is sitting at some corner of a street, unwashed, unshaved, left by his wife and family, switching two envelopes again and again, murmuring “it should work, it should work, my argument is airtight, I will be rich, I will be rich…”.

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.

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Posted: 22 November 2012 12:39 AM   [ Ignore ]   [ # 1664 ]
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GdB - 21 November 2012 11:35 AM

Clearer as here I cannot explain it, ...

Well, maybe I can. Don’t ever forget (if you do not agree with this then you are completely lost): the chance of an event is the number of times the event occurs, divided by the total number of events. Take the coin: there are two possible events, heads or tails. Say I throw 100 times, then we expect about 50 times heads. So the chance of heads is 50/100 = 0.5. And the die: the chance for a 3 is 1/6 means that if I throw the die 600 times, about 100 times it will be a 3. In both cases, based on symmetry I can say that the chances are effective 1/2 and 1/6 respectively. Right? If you do not agree with this definition of chance we are done.

Now the TEP: we must find the events we look for (maximise gain). Now it is a bit difficult to do this for all possible TEP pairs (these are infinite). But it suffices that the player does not know with what TEP pair he is playing. But we can probe the different possibilities for just one pair, and then state that this is true then for every TEP pair.

So what we must do is run through all possible scenarios for one TEP pair, with the addition that the player forgets immediate with what pair he is playing (or we play everytime with somebody else who does not know what has happened before).

Now we must compare the strategy ‘always switch’ with ‘not switching’.

As example we take the pair (10,20).

So the possible events are:
- You see you have chosen 10: you suppose the pair could have been (5,10) or (10,20). You switch, and you gain 10.
- You see you have chosen 20: you suppose the pair could have been (10,20) or (20,40). You switch, and you loose 10.

And not switching:
- You see you have chosen 10, you keep it, you loose 10.
- You see you have chosen 20, you keep it, you gained 10.

In both case the nett gain is 0.

Your rephrasing of the problem just gives you the wrong suggestion. Let’s compare it with the other situation, you choose an envelope, you look at the amount, and then the other envelope is filled with half or twice the amount you have, decided by the tossing of a fair coin (i.e. the chance of half or twice is 50%)

- You choose 10, you switch, the other contains 5, you loose 5
- You choose 10, you switch, the other contains 20, you gain 10 (from here I could stop, but for completeness I’ll do 20 too)
- You choose 20, you switch, the other contains 10, you loose 10
- You choose 20, you switch, the other contains 40, you gain 20

Obviously, you gain by switching. And obviously both situations are completely different. Clear now?

Corrolarium: the same argumentation holds if you do not look, so also in TEP without looking you do not gain by switching.

[ Edited: 22 November 2012 01:24 AM by GdB ]
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Posted: 22 November 2012 01:06 AM   [ Ignore ]   [ # 1665 ]
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GdB - 22 November 2012 12:39 AM

Now we must compare the strategy ‘always switch’ with ‘not switching’.

Yes, you are saying we can only take a general view.

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.

Stephen

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