The Bent Coin

[wtfsvi]

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A few people, however, will tell you that they are equally likely.

[/ QUOTE ] They are not equally likely. We know for certain that these two outcomes are not equally likely, because the coin is bent. But from our point of view we don't know which outcome is more likely, so to us both outcomes are equally likely.

As soon as you introduce another factor to this, like you tell me you're willing to bet on heads, my point of view changes and now I have reason to believe heads is more likely than tails.

[RJT]

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The only real contention to your OP is that the real world is never modeled by your conditions. Or that where it is, it is impossible or extremely difficult to combine probability and non probability considerations.

Which is a very interesting debate that we should be having instead.

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Isn’t that what we have been doing, for the most part, here on SMP all along?

[jason1990]

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Your suggestion wasn't ...

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My only suggestion was this:

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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.

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That is the inspiration for this thread. This might say it all:

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I think the only arguments in this thread that aren't trivial are the ones that are trying to show that the rest of the thread is trivial.

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[T50_Omaha8]

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This might say it all:

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I think the only arguments in this thread that aren't trivial are the ones that are trying to show that the rest of the thread is trivial.

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[/ QUOTE ]lol...what does my antagonistic (towards you) line have to do with anything? I think you're wrong and you're trying to weasel your way out of it. The only point you have is on an obvious and trivial technicality.

[jason1990]

If you want to talk about probabilities as seen from your particular point of view, then go right ahead. A lot of people do that. There is a rich tradition in probability theory of doing exactly that. But some people with your point of view like to think that it represents some objective reality, that it has some scientific foundations. Such a claim, of course, is trivially absurd. The point of view that (to you) the outcomes are equally likely is based on a total lack of evidence. Any conclusions drawn from this point of view are likewise based on a total lack of evidence. Scientific claims are based on evidence.

[jason1990]

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This might say it all:

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I think the only arguments in this thread that aren't trivial are the ones that are trying to show that the rest of the thread is trivial.

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[/ QUOTE ]lol...what does my antagonistic (towards you) line have to do with anything?

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It applies because it is true. My original claim, as described by David in the OP, is correct, obvious, and trivial. The rest of David's OP has nothing to do with my original claim.

[PairTheBoard]

A problem here is that Sklansky has not told us exactly what he means by the "probability" the coin lands heads. He has not described a controlled experiment which we can repeat and measure frequencies to test if his idea of probability makes any sense to us. The reason this example is important is that David often does this kind of thing and people are not always aware of the problems hiding in its ambiguity. Jason has provided a probalility model below where the conclusions of the model are mathematically trivial. But it also illustrates the important point that the correct description of the 50-50 chance in the model is not the same as the vague description Sklansky uses. Furthermore, there ia an important practical point that if we settle for the vague language Sklansky often uses for his theories we can be easily mislead into error.

Jason's Mathematical Model
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My original point here is very simple. There is a bent coin. Consider these two events:

A1: the coin lands heads
A2: the coin lands tails

These events are not equally likely. Any reasonable person should admit that, and any reasonable person should tell you that you cannot determine the probabilities of these events without doing further analysis on the coin.

A few people, however, will tell you that they are equally likely. They use their lack of knowledge to assume they are equally likely, and then present that assumption as some sort of empirical fact. It is simple to see their error in the bent coin example. It is harder to see their error in more complicated examples.

David has shifted the focus away from A1 and A2. He has introduced:

B1: David's fair coin and my bent coin land on the same face.
B2: David's fair coin and my bent coin land on opposite faces.

These events are equally likely. Any reasonable person should admit this. And anyone with basic probability knowledge should be able to prove that this follows from the fairness of David's coin. If we have good, practical reasons to believe that David's coin is fair, then there is nothing to discuss. It is trivial.

David breaks even betting on B1 or B2, but this has nothing to do with my original point.
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The important practical point
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any event with two possible outcomes that we know nothing about the probability of, can be assumed to have a 50% chance of each outcome.


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This is exactly the kind of thinking my comments are meant to correct. Of course you can assume anything you want. That does not mean you are right. You assume this at your own peril. If you are going to assume that two things are equally likely, then you should have a good reason for assuming this, especially if you are going to act on that assumption in any significant way.

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Futhermore, I don't think the repeatable experiment David really has in mind is the one Jason described above. I believe David thinks he can really just choose Heads and have a 50% one time probability of being right. What he has in mind is an imaginary world full of people who have bent coins in their pockets. He chooses one of these people at random, picks Heads, has them take the coin out and flip it. Maybe he bets on it with his buddy rather than the person with the coin.

His hidden assumption then is that coins get bent randomly and are equally likely to be bent favoring heads as tails. Under the assumptions of that model it makes sense to say that a randomly chosen person with a randomly bent 2 sided coin has a 50% chance of being flipped heads. Under those assumptions we could theoretically repeat the experiment and test out their validity. The problem is that it's an imaginary scenario that can't really be tested. Jason's model is one that can be tested. All that it reqires is the one bent coin.

So David's model has a lot of hidden assumptions in an imaginary world where the assumptions can't easily be tested. In my opinion, when David asserts theories involving probabilities for 1 time events it behooves us to ask him what his model and underlying assumptions are. They might not always be realistic. Can they be tested or is he just pronouncing them. Until I can take a look at them I'm not going to just take his word for it that he's making any sense.

PairTheBoard

[David Sklansky]

"A few people, however, will tell you that they are equally likely. They use their lack of knowledge to assume they are equally likely, and then present that assumption as some sort of empirical fact. It is simple to see their error in the bent coin example. It is harder to see their error in more complicated examples."

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.

[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 "two choices"? Flip a coin and pick one? Now look at Jason's model.

PairTheBoard

[David Sklansky]

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.