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From: Jesse Mazer <lasermazer.domain.name.hidden>

Date: Wed, 06 Oct 2004 15:26:48 -0400

Stathis Papaioannou wrote:

*>The problem is that you are reasoning as if the amount in each envelope can
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*>vary during the game, whereas in fact it is fixed. Suppose envelope A
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*>contains $100 and envelope B contains $50. You open A, see the $100, and
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*>then reason that B may contain either $50 or $200, each being equally
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*>likely. In fact, B cannot contain $200, even though you don't know this
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*>yet. It is easy enough for an external observer (who does know the contents
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*>of each envelope) to calculate the probabilities: if you keep the first
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*>envelope, your expected gain is 0.5*$100 + 0.5*$50 = $75. If you switch,
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*>your expected gain is 0.5*$100 (if you open B first) + 0.5*$50 (if you
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*>open A first) = $75, as before.
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*>
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*>Ignorance of the actual amounts may lead you to speculate that one of the
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*>envelopes may contain $200, but it won't make the money magically
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*>materialise! And even if you don't know the actual amounts, the above
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*>analysis should convince you that nothing is to be gained by switching
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*>envelopes.
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*>
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*>If the game changes so that, once you have opened the first envelope, the
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*>millionaire decides by flipping a coin whether he will put half or double
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*>that amount in the second envelope, then you are actually better off
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*>switching.
*

I don't think that's a good counterargument, because the whole concept of

probability is based on ignorance--if you were omniscient, for example, you

wouldn't have a need for probabilities at all. If someone puts $1000 in a

blue envelope and then flips a coin to decide whether to put $3000 in a red

envelope or to leave it empty, my expected gain from picking the red

envelope should be $1500 dollars--it doesn't make sense to say that from the

point of view of someone who saw the envelopes being stuffed, it is already

certain whether the red envelope contains the money, therefore *my* expected

gain from picking the red envelope should be either $3000 or zero. My

expected gain is based on my own ignorance of the outcome of the coin toss,

information that I don't have access to shouldn't play a role. Similarly, an

external observer who knows the content of both envelopes should play no

role in the calculation of my expected gain from switching in the

two-envelope problem. If I open the envelope and find $50, I don't know

whether the other envelope contains $25 or $100, so that information cannot

be used when calculating my expected gain from switching. But as I argued

before, if I know the probability distribution the envelope-stuffer used to

pick the amount in the envelope with less money, then seeing the amount in

the envelope I open will allow me to refine my estimate of the probability

it's the envelope with less money, there's no possible distribution the

envelope-stuffer could use that would insure that no matter how much I found

in the first envelope, the other envelope would have a 50% chance of

containing double that and a 50% chance of containing half that.

Jesse

Received on Wed Oct 06 2004 - 15:39:15 PDT

Date: Wed, 06 Oct 2004 15:26:48 -0400

Stathis Papaioannou wrote:

I don't think that's a good counterargument, because the whole concept of

probability is based on ignorance--if you were omniscient, for example, you

wouldn't have a need for probabilities at all. If someone puts $1000 in a

blue envelope and then flips a coin to decide whether to put $3000 in a red

envelope or to leave it empty, my expected gain from picking the red

envelope should be $1500 dollars--it doesn't make sense to say that from the

point of view of someone who saw the envelopes being stuffed, it is already

certain whether the red envelope contains the money, therefore *my* expected

gain from picking the red envelope should be either $3000 or zero. My

expected gain is based on my own ignorance of the outcome of the coin toss,

information that I don't have access to shouldn't play a role. Similarly, an

external observer who knows the content of both envelopes should play no

role in the calculation of my expected gain from switching in the

two-envelope problem. If I open the envelope and find $50, I don't know

whether the other envelope contains $25 or $100, so that information cannot

be used when calculating my expected gain from switching. But as I argued

before, if I know the probability distribution the envelope-stuffer used to

pick the amount in the envelope with less money, then seeing the amount in

the envelope I open will allow me to refine my estimate of the probability

it's the envelope with less money, there's no possible distribution the

envelope-stuffer could use that would insure that no matter how much I found

in the first envelope, the other envelope would have a 50% chance of

containing double that and a 50% chance of containing half that.

Jesse

Received on Wed Oct 06 2004 - 15:39:15 PDT

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