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Poker Forums => The Rail => Topic started by: MrsLime on October 09, 2006, 12:25:48 AM



Title: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 12:25:48 AM
If you liked the 'Monty Hall' problem, you're going to hate this one!

Imagine I have a sum of money and two envelopes.  I put one-third of the money into one envelope, and two-thirds of the money into the other envelope (i.e. one envelope contains twice as much as the other).  I give one envelope to Snoopy and one to Danafish, and tell them that before they look inside, they can either keep their envelopes, or swap them with each other.

Now, Snoopy thinks to himself: "Imagine my envelope contains £20.  Then there is a 50% chance that Danafish's envelope contains £10, and 50% that Danafish's envelope contains £40.  So if I swap envelopes, 50% of the time I lose £10 and 50% I gain £20.  So, on average, I gain £5 by swapping.  Therefore I should swap!"

At the same time, Danafish is thinking to herself: "Imagine my envelope contains £20.  Then there is a 50% chance that Snoopy's envelope contains £10, and 50% that Snoopy's envelope contains £40.  So if I swap envelopes, 50% of the time I lose £10 and 50% I gain £20.  So, on average, I gain £5 by swapping.  Therefore I should swap!"

But of course there is only a fixed sum of money between the two envelopes, so how is it possible that they both gain by swapping?


Title: Re: So you thought Monty Hall was tricky?
Post by: londonpokergirl on October 09, 2006, 12:27:33 AM
Not this one again :)   let me remember.....



Title: Re: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 12:30:51 AM
Ok, anyone from the Slough Palace can ignore this thread!


Title: Re: So you thought Monty Hall was tricky?
Post by: AndrewT on October 09, 2006, 12:36:00 AM
I thought it must be something to do with the phrasing of the question, but it isn't.

I had a think, then gave up and googled the answer.

Then my brain melted.


Title: Re: So you thought Monty Hall was tricky?
Post by: londonpokergirl on October 09, 2006, 12:38:34 AM
Ok, anyone from the Slough Palace can ignore this thread!
hehe me and patrick out then :)


Title: Re: So you thought Monty Hall was tricky?
Post by: Mbuna on October 09, 2006, 12:47:55 AM
Both players win as they have risked 0 to win a either 1/3 of a sum of money or 2/3 of a sum of money .

50% of the time a player will win 2/3 of the money regardless of wether they swap or not
and 50% he will win 1/3.

I think


Title: Re: So you thought Monty Hall was tricky?
Post by: Mbuna on October 09, 2006, 12:49:51 AM
I personally would swap if my envelope was the thinest.


Title: Re: So you thought Monty Hall was tricky?
Post by: Colchester Kev on October 09, 2006, 12:51:08 AM
If some mug was giving away free money in envelopes, i would cut a deal with the other envelope holder and do an even chop :)


Title: Re: So you thought Monty Hall was tricky?
Post by: thetank on October 09, 2006, 05:47:56 AM
The envelope with a third has 2/6 of the dosh
The envelope with two thirds has 4/6 of the dosh

Equal chance of having both, so your EV before the envelope opens is a half of 2/6 + 4/6. This is 3/6.

If you swich and lose, you drop 1/6 from your EV, if you switch and win, you're up 1/6 from your EV.

As 1/6 - 1/6 = 0 neither player will either gain or lose from swapping.

....but Snoopy is a jammy fish, so I'd probably swap if I was Dana.  :D


Title: Re: So you thought Monty Hall was tricky?
Post by: thetank on October 09, 2006, 06:17:33 AM
But that's all common sense of course. Everyone knows that neither player gets anything out of swapping.
So one can conclude, as AndrewT said,  there must be a logical fallacy in the wording.

Here is my theory.....


Imagine my envelope contains £20.


This assumes that there is a 100% chance of his envelope containing £20 (or a scalar multiple thereof)
In truth, there's a 50% chance it has £20, and a 50% chance it has another amount. (25% £10, and 25% £40)

So before you know what's in the envelope it is not only £20, it is also £10, and also £40. Something to do with a cat belonging to Schroedinger..blah blah blah, but it's at this point that the logical fallacy draws it's root, and skews the sum.

 


Title: Re: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 02:59:07 PM
Here is my theory.....

Imagine my envelope contains £20.

This assumes that there is a 100% chance of his envelope containing £20 (or a scalar multiple thereof)
In truth, there's a 50% chance it has £20, and a 50% chance it has another amount. (25% £10, and 25% £40)

So before you know what's in the envelope it is not only £20, it is also £10, and also £40. Something to do with a cat belonging to Schroedinger..blah blah blah, but it's at this point that the logical fallacy draws it's root, and skews the sum.

I think perhaps you've misunderstood what 'imagine' means and have been diverted by thoughts of kittens barking up the wrong tree.  Let me slightly reword the third paragraph:

Now, Snoopy thinks to himself: "I define X to be the amount in my envelope.  [So now, by definition, there is 100% chance that the envelope contains X.]  Then there is a 50% chance that Danafish's envelope contains X/2, and 50% that Danafish's envelope contains 2X.  So if I swap envelopes, 50% of the time I lose X/2 and 50% I gain X.  So, on average, I gain X/4 by swapping.  Therefore I should swap!"

Of course we agree that there can be no value in swapping!  But where is the logical flaw in Snoopy's reasoning?


Title: Re: So you thought Monty Hall was tricky?
Post by: AndrewT on October 09, 2006, 03:08:18 PM
Here is my theory.....

Imagine my envelope contains £20.

This assumes that there is a 100% chance of his envelope containing £20 (or a scalar multiple thereof)
In truth, there's a 50% chance it has £20, and a 50% chance it has another amount. (25% £10, and 25% £40)

So before you know what's in the envelope it is not only £20, it is also £10, and also £40. Something to do with a cat belonging to Schroedinger..blah blah blah, but it's at this point that the logical fallacy draws it's root, and skews the sum.

I think perhaps you've misunderstood what 'imagine' means and have been diverted by thoughts of kittens barking up the wrong tree.  Let me slightly reword the third paragraph:

Now, Snoopy thinks to himself: "I define X to be the amount in my envelope.  [So now, by definition, there is 100% chance that the envelope contains X.]  Then there is a 50% chance that Danafish's envelope contains X/2, and 50% that Danafish's envelope contains 2X.  So if I swap envelopes, 50% of the time I lose X/2 and 50% I gain X.  So, on average, I gain X/4 by swapping.  Therefore I should swap!"

Of course we agree that there can be no value in swapping!  But where is the logical flaw in Snoopy's reasoning?

The problem is that you have defined X to be the amount in Snoopy's envelope. Then you take two separate cases, one where Snoopy has the smaller amount and one where he has the larger amount. You then equate the two values of X, which you can't do because they're different.

The X in 2X is a different X than the X in X/2.


Title: Re: So you thought Monty Hall was tricky?
Post by: Mbuna on October 09, 2006, 03:30:57 PM
Here is my theory.....

Imagine my envelope contains £20.

This assumes that there is a 100% chance of his envelope containing £20 (or a scalar multiple thereof)
In truth, there's a 50% chance it has £20, and a 50% chance it has another amount. (25% £10, and 25% £40)

So before you know what's in the envelope it is not only £20, it is also £10, and also £40. Something to do with a cat belonging to Schroedinger..blah blah blah, but it's at this point that the logical fallacy draws it's root, and skews the sum.

I think perhaps you've misunderstood what 'imagine' means and have been diverted by thoughts of kittens barking up the wrong tree.  Let me slightly reword the third paragraph:

Now, Snoopy thinks to himself: "I define X to be the amount in my envelope.  [So now, by definition, there is 100% chance that the envelope contains X.]  Then there is a 50% chance that Danafish's envelope contains X/2, and 50% that Danafish's envelope contains 2X.  So if I swap envelopes, 50% of the time I lose X/2 and 50% I gain X.  So, on average, I gain X/4 by swapping.  Therefore I should swap!"

Of course we agree that there can be no value in swapping!  But where is the logical flaw in Snoopy's reasoning?

The problem is that you have defined X to be the amount in Snoopy's envelope. Then you take two separate cases, one where Snoopy has the smaller amount and one where he has the larger amount. You then equate the two values of X, which you can't do because they're different.

The X in 2X is a different X than the X in X/2.
;iagree;

To put it another way the envelope contains either the larger amount in (say 2X) if we swap we lose X

OR

The envelope contains the smaller amount ie X in which case if we swap we gain X


50% of the time we gain X 50% we lose X

 EV+ FROM THE SWAP = 0



Title: Re: So you thought Monty Hall was tricky?
Post by: ItsMrAlex2u on October 09, 2006, 03:40:00 PM
My brain is hurting now !!


Title: Re: So you thought Monty Hall was tricky?
Post by: thetank on October 09, 2006, 04:09:06 PM

The problem is that you have defined X to be the amount in Snoopy's envelope. Then you take two separate cases, one where Snoopy has the smaller amount and one where he has the larger amount. You then equate the two values of X, which you can't do because they're different.

The X in 2X is a different X than the X in X/2.


That sounds more likely.


Title: Re: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 05:52:11 PM
Ok, maybe algebra was a bad idea.  How about:

Snoopy opens the envelope and finds £20.  I.e. the envelope definitely contains £20!  He thinks to himself: "Great I have £20, but... there is a 50% chance that Danafish's envelope contains £10, and 50% that Danafish's envelope contains £40.  So if I swap envelopes, 50% of the time I lose £10 and 50% I gain £20.  So, on average, I gain £5 by swapping.  Therefore I should swap!"

Now where is the flaw?

(BTW, I don't know the answer, I am trying to understand as well.)


Title: Re: So you thought Monty Hall was tricky?
Post by: roverthtaeh on October 09, 2006, 06:07:01 PM
Both people think they gain by swapping. Agreed.
Here's my philosophy:
The other guy wants to swap because he thinks he'll gain.
So I refuse to swap, therefore I gain what he thought he would gain.
And if I'm wrong, he goes "neener neener" and sticks his tongue out.
But I'm financially better off than I was 5 minutes ago, so who cares.


Title: Re: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 06:15:43 PM
Both people think they gain by swapping. Agreed.
Here's my philosophy:
The other guy wants to swap because he thinks he'll gain.
So I refuse to swap, therefore I gain what he thought he would gain.

Unfortunately, your logic is far from impeccable.

Suppose you say to me: "Do you want to swap this used tampon for $1,000,000?"
I say "No."
Therefore I have just gained $1,000,000?


Title: Re: So you thought Monty Hall was tricky?
Post by: roverthtaeh on October 09, 2006, 06:20:28 PM
You under-rate used tampons


Title: Re: So you thought Monty Hall was tricky?
Post by: Mbuna on October 09, 2006, 06:21:28 PM
No you havnt gained 1Million you have NOT LOST 1million  BIG Difference


Title: Re: So you thought Monty Hall was tricky?
Post by: roverthtaeh on October 09, 2006, 06:27:20 PM
Yeah, and I've still got my tampon


Title: Re: So you thought Monty Hall was tricky?
Post by: MrsLime on October 09, 2006, 06:30:53 PM
You under-rate used tampons

Hey, I value used tampons as much as the next man, but I doubt you could present me with one so juicy and pungent that it was worth ONE MILLION DOLLARS.  Now you are just being silly.


Title: Re: So you thought Monty Hall was tricky?
Post by: roverthtaeh on October 09, 2006, 06:43:04 PM
You under-rate used tampons

Hey, I value used tampons as much as the next man, but I doubt you could present me with one so juicy and pungent that it was worth ONE MILLION DOLLARS.  Now you are just being silly.

What might one be worth from the private suite of, say... Angelina Jolie?
And could I try it on ebay?


Title: Re: So you thought Monty Hall was tricky?
Post by: Claw75 on October 09, 2006, 07:31:38 PM
this thread has turned gross!


Title: Re: So you thought Monty Hall was tricky?
Post by: byronkincaid on October 09, 2006, 07:43:43 PM
Quote
Hey, I value used tampons as much as the next man

All my fantasies dashed... Oh well there's always that new character in Lost, Juliet  ;kev;