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#11
08-08-2006, 04:25 PM
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Re: Envelopes
[ QUOTE ]
I think its pretty normal to first, realize the obvious answer is 'switch,' and second, be confused by how it seems an obvious +EV move to switch, yet choosing randomly should be the same EV...right? This is called the Monty Hall problem, with a linky here [/ QUOTE ] it's not the monty hall problem. 
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#12
08-08-2006, 04:49 PM
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Re: Envelopes
[ QUOTE ]
[ QUOTE ] I think its pretty normal to first, realize the obvious answer is 'switch,' and second, be confused by how it seems an obvious +EV move to switch, yet choosing randomly should be the same EV...right? This is called the Monty Hall problem, with a linky here [/ QUOTE ] it's not the monty hall problem. [/ QUOTE ] Its very similar. Its the counterintuitive idea that probabilites or outcomes can change when no new information is added. I guess it is a little different, but I think the source of the incredulity is very similar.
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#13
08-08-2006, 04:56 PM
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Re: Envelopes
[ QUOTE ]
Its very similar. Its the counterintuitive idea that probabilites or outcomes can change when no new information is added. [/ QUOTE ] IIRC in the monty hall problem you do get more information and you do improve your situation by switching.
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#14
08-08-2006, 05:18 PM
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Re: Envelopes
This is not the Monty Hall problem. There's a very long thread in probablility about this that I'm too lazy to link for you. Summary:
Always switching does not change your EV vs. never switching. In order to make the estimate of 50% of your amount being the big one vs. 50% of your amount being the small one, you have to have some estimate of the distribution of amounts. By assuming that, no matter the number, you're equally likely to have the small number as the big, you're assuming that any amount is equally likely. That's obviously false. $1 billion? $100 billion? $1 googol? At the minimum, this paradox can be escaped by guessing the largest amount that you might be given, say $1 million. In that case, you maximize EV by keeping $500,000 or more and switching anything else. I don't have enough probability to get deep into it, but the value of this paradox is making you think critically about your preconceptions (known in stats as "priors") about what you expect to see in the envelope. Most people reflexively assuming something irrational, that any number is equally likely.
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#15
08-08-2006, 05:40 PM
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Re: Envelopes
This reminded me of this problem:
You have a black box with a slot in the front and a button. There are an unknown number of marbles in the box, one of which is white, all the rest are black. The box is magic, in that the number of balls that may fit in the box does not depend on its physical size. Each time you press the button, the box spits out a marble, chosen at random from the marbles within. You push the button N times, and on the Nth attempt, the white marble is ejected. About how many marbles would you say are in the box, why, and how certain are you?
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#17
08-08-2006, 06:38 PM
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Re: Envelopes
[ QUOTE ]
I throw away my envelope and punch you in the nose for wasting my time for a maximum payoff of $20. [/ QUOTE ] It probably took you as long to type that post as it takes to pick an envelope. So, how much did you get paid for it? $100? $200?
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#19
08-08-2006, 09:34 PM
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Re: Envelopes
[ QUOTE ]
Most people reflexively assuming something irrational, that any number is equally likely. [/ QUOTE ] It's not so much that it's irrational; it's that it's IMPOSSIBLE for that to be the case. That's why the "always switching is optimal" argument fails. There are a number of threads on this problem. Search for them.
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#20
08-08-2006, 10:44 PM
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Re: Envelopes
This is wrong. Take a look at this:
Scenario 1: Envelop A contains $5 and B contains $10 (but you DON'T know this!). a) You play 1,000,000 times, picking one at random, switching after seeing what's in there. Assuming 500,000 for each envelop, you win a total of $7,500,000, for an EV of $7.5. b) Now you do the same again, but without switching. The result is again an EV of $7.5, since you´re still getting envelops A and B 500,000 times each, only you get them straight up instead of after switching. Scenario 2: Envelop A contains $10 and B contains $20. a) switching after each pick, your EV is $15. b) not switching, your EV is $15 again. As others stated, looking at the envelop doesn´t reveal any useful information, which is not the case in monty hall's problem.
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