The Bent Coin

[PairTheBoard]

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Forget the flipping coins to make your pick. I used that only to avoid taking the worst of it if the other guy can choose how many times to bet.

Meanwhile I have no idea what your post means.

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I could say the same about yours here,

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Probability does not apply to real objects. It applies to the EVIDENCE OR INFORMATION ABOUT THE OBJECT and the experiment. The frequency distribution regarding that evidence. 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. THAT is harder to see in more complicated examples.

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What 2 choices? Let's call them A and B. We have no other information about them? One or both or neither might be true? Or did you mean they were mutually exclusive and exhastive? In other words, B=Not A? To simplify, let's assume one is true and one is false, but we don't know which. Do you not see the difference between these two statements,

1. If I flip a fair coin to choose, I have a 50% probability of choosing the true statement.

2. A has a 50% probability of being true.

Why is it so important to understand the difference between those two statements? Well, let's start looking at the implications of #2. Suppose we do a little logical work and based on nothing else but logical trivialities, ie. no empirical evidence, we determine the following logical chain of equivalent statements.

A <==> A1 <==> A2 <==> A3 <==> (2+2=5)

Do you now assert that there is a 50% probability that the statement (2+2=5) is true?

Of course, you can say that we now have more information about A. We have deduced some trivial logical equivalences. But has that really introduced new information? That information was logically contained in A to begin with. We added no empirical evidence.

Still not convinced? Suppose instead of arriving at (2+2=5) we instead arrive at a less obviously incorrect statement. Suppose we arrive at a statement that only makes us mildly uncomfortable, but we cannot pinpoint the discomfort nor is there any real life situation we can apply to gain empirical data one way or the other. Despite our discomfort Sklansky can stand on the original #2 statement he has forced us to accept and now assert that this statement of discomforture must have a 50% probability of being true. Do you see anything wrong with that yet?

Of course there's only the claim of 50% probability so far, and only mild discomfort. But suppose Sklansky can parlay a number of these statements. Suppose he has statements A1,B1; A2,B2; A3,B3, ..., A7,B7 all pairs respectively comparitive to our original A,B. In other words, each Ai and Bi have a 50% probability of being true. And in each case, Ai is true if and only if Bi is false. Suppose further that there is no logical dependence between the Ai. In other words, Ai provides no information about when or under what conditions Aj might be true, and this holds for any combination of the Ai's as well. To make this more clear, all pairs of statements like,

(A1 and A2), [(B1 and B2) or (A1 and B2) or (A2 and B1)]

satisfy the same conditions as our original A,B. Furthermore, all pairs of statements like the above involving any mutually exhastive combinations of any number of the Ai, Bi satisfy the conditions of our original A,B.

Now suppose we find the trivial logical chains,

A1 <==> ... <==> (Extremely uncomfortable conclusion 1)
A2 <==> ... <==> (Extremely uncomfortable conclusion 2)
.
.
.
A7 <==> ... <==> (Extremely uncomfortable conclusion 7)

Once again, none of these EUC's 1 thru 7 are obvious logical falsehoods like (2+2=5), nor can any of them be empirically tested. Nor do any of them have obvious relationships to any known probabilities. Yet they involve concepts whereby we intuitely resist the conclusions even though we can't produce any hard evidence to justify our resistance to them. They just don't pass our sniff test. None of them.

Yet by Sklansky-Probability-Logic we are forced to listen to his conclusion that there is a 99% probability that one of them must be true.

Are you starting to get an idea for why we might want to stick to the Mathematically sound statement #1 instead of the vague mathematically undefined statement #2?

PairTheBoard

[PairTheBoard]

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Probability does not apply to real objects. It applies to the EVIDENCE OR INFORMATION ABOUT THE OBJECT and the experiment. The frequency distribution regarding that evidence. 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. THAT is harder to see in more complicated examples.

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What 2 choices? Let's call them A and B. We have no other information about them? One or both or neither might be true? Or did you mean they were mutually exclusive and exhastive? In other words, B=Not A? To simplify, let's assume one is true and one is false, but we don't know which. Do you not see the difference between these two statements,

1. If I flip a fair coin to choose, I have a 50% probability of choosing the true statement.

2. A has a 50% probability of being true.

Why is it so important to understand the difference between those two statements?

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Still not convinced from my last post. Try this one. Consider the example of the Two Envelope problem. There are two sealed Envelopes. You are told that one Envelope has twice the amount of money in it than the other. You are given no other information. You have no information about how the Envelope amounts were chosen.

The two Envelopes are randomly suffled. You are told you can pick one and keep the amount of money in the envelope. Or, after looking inside the envelope you can choose to pay 10% of the amount in the envelope, switch and take the amount in the second envelope.

You pick an envelope, look inside and see $100. You ask yourself, should I pay $10 so I can switch to the other envelope? You think to yourself, the other envelope must have either $200 or $50 in it. You think, I have no other information than that. Is switching correct or is it a mistake. If I use the mathematically correct principle #1 above and flip a coin to decide whether to switch or stand pat, then I know I will have a 50% chance of being correct. But that doesn't tell me whether the coin is letting me correctly switch to a $200 envelope half the time, or correctly stand pat half the time vis a vie $50 in the other envelope. So #1 doesn't help me decide whether I should pay $10 to switch.

Ah! I know. I'll apply Sklansky-Probability-Logic. Either there is $200 in the other envelope or $50. One or the other. I don't know which and I have no other information about how the envelope amounts were chosen. That means I can apply the indifference principle and conclude there is a 50% chance of the $200 and a 50% chance of the $50. hmmmm, let's see. (.5)(200) + (.5)(50) = $125. Hey. 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. That GREAT! I'm paying the $10 to make the switch.

Still think #2 is such a slam dunk? DUCY yet?

PairTheBoard

[jason1990]

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Probability does not apply to real objects.

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I did not know you felt this way. I know several scientists that would disagree with this sentence.

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It applies to the EVIDENCE OR INFORMATION ABOUT THE OBJECT and the experiment. The frequency distribution regarding that evidence. 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. THAT is harder to see in more complicated examples.

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What are you talking about? Do you really think the history of the universe has something to do with this? Okay, look, I just bent a real penny. It is sitting here on my desk. (Do you think this penny is affected by the history of the universe?) Here are two questions:

(a) What is the probability the penny comes up heads?
(b) What is the probability the penny comes up tails?

Pick a question, either question, and answer it. You can answer with "not enough information" or you can answer with a number between 0 and 1.

All of your posts seem to indicate that your answer to both questions is 0.5. Yet I do not think you have come right out and said that. Why? Am I misunderstanding you?

[SNOWBALL]

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.

[blah_blah]

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Probability does not apply to real objects.

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I did not know you felt this way. I know several scientists that would disagree with this sentence.

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this seems like a particularly odd view for someone who expresses an essentially Bayesian view of probability as does David.

[Phil153]

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Okay, look, I just bent a real penny. It is sitting here on my desk. (Do you think this penny is affected by the history of the universe?) Here are two questions:

(a) What is the probability the penny comes up heads?
(b) What is the probability the penny comes up tails?

Pick a question, either question, and answer it. You can answer with "not enough information" or you can answer with a number between 0 and 1.

All of your posts seem to indicate that your answer to both questions is 0.5. Yet I do not think you have come right out and said that. Why? Am I misunderstanding you?

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Here are my answers:

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

Two questions for you:

(a) What is the best probability estimate we can assign to heads?
(b) What is the best probability estimate we can assign to tails?

You can answer "not enough information", or a number between 0 and 1.

You may think these best probability estimates are useless, but they're not. To see why, consider this question:

- Someone loads a dice to fall a certain way more often than normal. He offers you 7 to 1 odds to pick a number and roll it. Do you take it?

- Someone loads a roulette wheel to land on a certain number more often than normal. He offers you 7 to 1 odds to pick a number and spin the wheel. Do you take it?

You can see that our imperfect best probability estimate (using all known information) is indeed useful. This is very hard to see and trust in more complex cases, so people dismiss it as they're doing in this thread.

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In your 'example', you aren't really assigning an estimate to either heads or tails, you're just noting that these is a randomized strategy that allows you to make +EV bets as long as EV of making randomized bet with given odds is > total payouts / number of choices.

The existence of such a strategy doesn't tell you anything about the probabilities of the coin landing heads or tails respectively.

Also the idea of 'best probability estimate' depends essentially on your definition of 'best', 'probability', and 'estimate' and there isn't really a canonical choice.

[PairTheBoard]

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In your 'example', you aren't really assigning an estimate to either heads or tails, you're just noting that these is a randomized strategy that allows you to make +EV bets as long as EV of making randomized bet with given odds is > total payouts / number of choices.

The existence of such a strategy doesn't tell you anything about the probabilities of the coin landing heads or tails respectively.

Also the idea of 'best probability estimate' depends essentially on your definition of 'best', 'probability', and 'estimate' and there isn't really a canonical choice.

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This is correct. For the die and roulette examples I use the same kind of randomizing method to make the bet. It says nothing about "estimating" probabilities for the weighted die or rigged wheel.

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

[blah_blah]

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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.

[jason1990]

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