The Bent Coin

[Piers]

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In this case it is talking about all situations where there are two choices and there is no other information. And in the history of the universe both logic and experiment would agree that the two choices come up equally often with that evidence.

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This is a extreemly dodgy statment. In particular the "two choices and there is no other information" condition. No other information? For example your two coins example would not satisfy this conditions, knowledge that you are tossing two coins is farily clearly furthur information.

[chezlaw]

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Look at my Two Envelope example above Phil to see the kind of trouble you can get into making your estimate of the probability based on the indifference principle.

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Isn't it indifference to labels and symmetry that matters here? Head, tails are arbitary labels, swapping the labels makes no difference to anything so we can be indifferent to them. Unlike the money in envelopes.

chez

[Piers]

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You have no information about how the Envelope amounts were chosen.

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This is the key, you have to form a probablity ditribution for the amounts in the envolpe. Is $50 more or less likely than $1000000000000000000000000000000000000000000000000 000000000?
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I'm on to something here. Switching is +EV for me. The other envelope is worth $25 more in EV than the one I've got

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The calculation you use assumes a uniform distribution across the positive numbers, however this does not form a valid porbality space. You have to pick a finite distribution, and as soon as you do this everyting makes sense. Amounts less than the the worlds gross anual domestic product are clearly more likley than other amounts so
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You have no information about how the Envelope amounts were chosen.

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is likely unbelievable, and does not equate to you have no informatio about how the envolope amounts are distrubuted.

Just explaining, probablily missing the deeper point.

[PairTheBoard]

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Look at my Two Envelope example above Phil to see the kind of trouble you can get into making your estimate of the probability based on the indifference principle.

PairTheBoard

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I view the Two Envelope 'paradox' to be more of a cautionary tale of what happens when naive ideas of probability (that imply switching indefinitely leads to an unbounded EV) are applied to problems that require a careful theoretical foundation (probability distributions, translation-invariance, etc).

The problem we are debating here (inasmuch as it is a clearly defined problem) doesn't really require any specialized knowledge of probability; everyone seems to agree that certain conclusions can be drawn which are essentially dependent on the expected value of the problem, but people are trying to deconstruct this to obtain some sort of a priori estimate on the individual probabilities, which is wishful thinking at best without some additional information.

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Exactly. It is wishful thinking. That's the whole point. The problem is people not realizing it's wishful thinking. And it's in the lack of that realization that they don't realize what they are doing in the Two Envelope problem is also wishful thinking that can lead them to the costly conclusion that they should pay the $10 to make the Envelope Switch. Had they become aware of the importance of realizing what they are doing in the bent coin example they might realize the need to look deeper into the two Envelope situation before paying their $10.

PairTheBoard

[Artsemis]

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2 doors. One of them has a prize, and the other is empty. Does anyone here deny that (absent other info) each door has a 50/50 probability of having the prize?

How about if you're the thousandth contestant on the show, and unbeknownst to you, the host ALWAYS leaves door #1 empty? From your perspective, the odds on door #1 are still 50/50, even though the audience knows that the odds on door #1 are actually 0% Probability is all all about perspective.



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This started off as a good example until you said "always". Instead, say there are two doors... one has a prize, the other doesn't.

The host knows door #1 is empty and door #2 has the prize. Your guess at door #1 is still 50%. It doesn't matter what the host knows (or what the penny-bender knows)... because next time the host may know it's the other door and that you are picking the correct one, but you are still at 50% on your guess.

[Phil153]

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Here are my answers:

(a) Not enough information
(b) Not enough information

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Then we agree. The rest of your post is arguing against something I never said.

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If your point is that we can't know the actual probability of a event shifted an unbounded, unquantifiable amount in an unknowable direction, then you haven't really given anything but tautology. Why didn't you just come out and say this in plain English to begin with, instead of giving bent coin examples?

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I did not say your "best guess" is useless. I said it is subjective.

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Our "best probability estimate" is subjective? I have to strongly disagree. If we're using the known information in a rational way to arrive at a number, it is not at all subjective. Possibly inaccurate in the light of further information - but not subjective.

[Beavis68]

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I don't think this discussion will have legs because I don't think jason's esoteric comments about probability interest many people here. But I do want to give a precise reply.

Jason says that if I tell you I bent a coin but don't show it to you, you cannot state the probability as to whether a flip will come up heads. If asked, the answer is "I don't have enough information."

I say that the information you have, in this case only the fact that there is two alternatives, allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose a side for the bent coin and get even money on your bets. You would win getting eleven to ten.

Any other definition of probability seems silly. Because you NEVER in real life have enough information. There is no fair coin.

There is more to be said but this will get things started.

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I believe you are right, but i think your variance would be higher depending on how the coin was bent.

For example if the coin was bent is such a way that it always came up one way, and you decided to stick with one call for 1000 flips, you will either be always right or always wrong.

Where as with a fair coin, you should be right around half the time.

[Beavis68]

I am not sure if anyone has done the math, but I think this is it.

Chance of the bent coin coming up heads = x chance of the bent coin coming up tails = 1-x.

Chance of you flipping heads with your coin when the bent coin is heads = 1/2 odds of flipping tails when the bent coin comes up tails = 1/2

The probability of the bent coin being heads and you flipping heads is x*1/2=1/2x

Probability of the bent coin coming up tails and you fillping tails on your coin = (1-x)*1/2 = 1/2 - 1/2x.

the probablility you will flip the correctly and win = 1/2x + (1/2 - 1/2x) = 1/2

What I got my brain around that made me accept this is that when you flip whatever why the coin is baised too, you will be correct much more often, so this even it out.

the bias (x) is not relavant to the equation

[MtDon]

David writes: [ QUOTE ]
I don't think this discussion will have legs because I don't think jason's esoteric comments about probability interest many people here. But I do want to give a precise reply.

Jason says that if I tell you I bent a coin but don't show it to you, you cannot state the probability as to whether a flip will come up heads. If asked, the answer is "I don't have enough information."

I say that the information you have, in this case only the fact that there is two alternatives, allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose a side for the bent coin and get even money on your bets. You would win getting eleven to ten.

Any other definition of probability seems silly. Because you NEVER in real life have enough information. There is no fair coin.

There is more to be said but this will get things started.

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I have to agree with Jason that the answer to the question as to what the probability is of the bent coin landing heads up is "I don't have enough information."

The bent coin does have a set probability of landing heads, for each specific way it is flipped. It is as much a physical property of the coin as its color for a given lighting condition.

You are right in saying that in the real world that there is no such thing as "a fair coin." Then phrase "a fair coin" as used in the mathematics of probability is an abstraction.

Jason's point, appears to me, is that it is best to keep one's thinking clear about the difference between what is actually known about a situation and what might be assumed about the situation which will sometimes lead to correct decisions.

Your writing that you could flip "a fair coin" (I assume before each flip of the bent coin) to determine whether to choose heads or tails implies that you also recognize that the bent coin doesn't actually have a 50 percent probability of coming up heads, but rather has an unknown probability of coming up heads. You are changing an unknown situation in to a random situation.

[NotReady]

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Yes. Sklansky should not have introduced this "fair" coin since it is determined to cause misunderstandings. I'm not quite sure what your misunderstanding is, so it's hard for me to address, but I'm quite sure you misunderstand.


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My apologies - I misread Sklansky's post, not sure why. He has an odd way of stating things and I saw something different than what he actually said - not the first time with one of his posts.