The Bent Coin

[David Sklansky]

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.

[chezlaw]

Don't we have to be careful what we are talking about the probability of?

If you get to call then you have a 50:50 chance of winning whether or not the coin is bent but that's not because the coin has a 50% chance of landing on heads or tails which is why you wouldn't let the guy who knows the coin call.

The reason you still have a fair chance if you call is because there's no disadvantage in not knowing which way the coin is unfair even if you know its unfair.

chez

[Phil153]

I don't understand the significance or relevance of Jason's coin.

Probability is about using the information you have available to you. If there is unknowable information, even if it changes your chance of winning to 100% or 0%, it in no way diminishes the probability model.

If he's trying to show that probability fails when there is non quantifiable information, a bent coin analogy isn't going to help.

Am I missing something?

[David Sklansky]

I should say at his point that I am not sure I have described Jason's contention correctly.

[ALawPoker]

I don't understand this either. Jason's claim seems absurdly semantical. It's like saying if you have AA vs. KK preflop that you don't have enough information to know if you're a favorite or not (maybe there's a 100% chance that a K will flop). But you have enough information to make the best decision available, which is all that matters.

[chezlaw]

[ QUOTE ]
I don't understand this either. Jason's claim seems absurdly semantical. It's like saying if you have AA vs. KK preflop that you don't have enough information to know if you're a favorite or not (maybe there's a 100% chance that a K will flop). But you have enough information to make the best decision available, which is all that matters.

[/ QUOTE ]
What's the probability of the zillionth digit of Pi being a three?

chez

[Phil153]

[ QUOTE ]
I should say at his point that I am not sure I have described Jason's contention correctly.

[/ QUOTE ]
I've finally read his post (I avoid esoteric stuff, prefer it easy) and his points are not clear, and where clear, they are not correct.

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.

[David Sklansky]

If you are trying to distinguish between events that have not yet happened and those that have, here is how I would undistinguish it. Assume the future event already happened and the two of you are gambling on a videotape. And what if neither one of you KNOW whether the KK vs AA hand is live or not?

[m_the0ry]

[ QUOTE ]
What's the probability of the zillionth digit of Pi being a three?

chez

[/ QUOTE ]

Depends on the numerical base, not enough information.

/images/graemlins/tongue.gif

[jason1990]

[ QUOTE ]
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.

[/ QUOTE ]
Based on the rest of your reply, what is your answer? 50%?

[ QUOTE ]
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.

[/ QUOTE ]
The same would be true if my bent coin was a two-headed coin. Are you suggesting a two-headed coin has probability 50% of coming up tails? Of course not. You are not really betting on the bent coin. You are betting on your fair coin.

The same would be true of anything you might bet on, in fact. You could flip your fair coin, if it comes heads you bet that the next roll on the craps table will be a 7, if it comes tails you bet that it will not be a 7. If someone pays you even money on your bets, you will break even. Do you really think this says something about the probability of rolling a 7? Of course not. You are not really betting on the roll of the dice. You are again betting on your fair coin.

The challenge is to

[ QUOTE ]
state the probability as to whether a flip will come up heads.

[/ QUOTE ]
You evade the question by using your fair coin to sometimes bet on tails. If you had to bet only on heads, what odds would you require? That is the question.

[ QUOTE ]
you NEVER in real life have enough information. There is no fair coin.

[/ QUOTE ]
Are you simply trying to point out that no coin is perfectly fair? If so, then I must say I do not find that topic interesting right now. My comments have so far been meant to address practical conclusions in ordinary circumstances. I have no present interest in discussing coin-flipping machines, coins that land on their edge, or the fact that the heads side of a quarter might be heavier than the tails side. Those discussions have their place, but they are not what I am talking about here.

A practical statistician will tell you that, under ordinary circumstances, it is reasonable to conclude that a typical unbent coin will land heads with probability 1/2. That same practical statistician will tell you that you cannot determine the probability of heads in a bent coin without further information. He would probably tell you that the best way to determine it is to flip the coin many times and use the results of those flips to estimate this probability.

This really is a very trivial observation on my part. I am astounded that you are so intent on debunking it.

[jason1990]

[ QUOTE ]
It's like saying if you have AA vs. KK preflop that you don't have enough information to know if you're a favorite or not (maybe there's a 100% chance that a K will flop).

[/ QUOTE ]
The reason you have enough information in this case is because the deck was shuffled. The shuffle may not be perfect, but as a practical matter it is usually reasonable to assume that it is, which justifies all of the ordinary probability calculations we do in cases like this.

But suppose you walk into a bar and see the AA and KK on the table, with the rest of the deck in the center. Someone offers to take the KK side and bet against you. I hope you would not accept this bet without first demanding to inspect the rest of the deck and then have it thoroughly shuffled.

When we assume that the next card is equally likely to be any of the unseen cards, we must have a reason to assume this. Complete ignorance is not a reason.

[chezlaw]

[ QUOTE ]
If you are trying to distinguish between events that have not yet happened and those that have, here is how I would undistinguish it. Assume the future event already happened and the two of you are gambling on a videotape. And what if neither one of you KNOW whether the KK vs AA hand is live or not?

[/ QUOTE ]
That's not the distinction I'm making. The zillionth digit in the (decimal) expansion is fixed in time but it's an unneccesary confusion to say there is a 10% chance of it being a three rather than if we guess unknown (to us) expansion digits we will get it right 10% of the time.

Unless you claim its not fixed until someone knows what it is. Some sort of probability field collapsing.

chez

[surftheiop]

DS- could this be a comparable analogy?
Arent you basically saying that if someone had a lottery ticket and said i took this ticket out of an unknown number of tickets with atleast one winning ticket in them, what is the chance it is winning. The coin could be bent in a way that could make the chance of flipping heads anywhere from 0-100% and in the ticket example the probabilty could either be 100% or in the range between aproximately 0-exactly 50%. I would have to say in both situations you dont have enough info to draw a guess on the probabilty

[wtfsvi]

[ QUOTE ]
The same would be true if my bent coin was a two-headed coin. Are you suggesting a two-headed coin has probability 50% of coming up tails? Of course not. You are not really betting on the bent coin. You are betting on your fair coin.

[/ QUOTE ] A two headed coin is not the same scenario as the OP. A coin that has two identical sides, but we don't know if they are heads or tails, is the same as the OP. And in such a scenario, we can assume a 50% chance of heads if we flip the coin.

edit: I agree, however, that using a fair coin to pick the bets is not good in pedagogical terms. There is no correlation between the facts that the fair coin is 50/50, and that we can assume that the bent coin is 50/50. A 11/10 bet on the bent coin will be +EV no matter what method we use to pick heads/tails. And 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.

[ALawPoker]

[ QUOTE ]
But suppose you walk into a bar and see the AA and KK on the table, with the rest of the deck in the center. Someone offers to take the KK side and bet against you. I hope you would not accept this bet without first demanding to inspect the rest of the deck and then have it thoroughly shuffled.

[/ QUOTE ]

If you offered me one of the sides, I'd of course assume that was the worse side if there was a chance you had more information than I did. Maybe David didn't set up your point correctly, in which case I'd correct him now, because it seems you are changing it slightly. In the coin example, the person who bent the coin does not pick a side. He lets you pick it, according to the OP.

It seems undeniable to me that I would still have a 50% chance of picking the winning side. In the AA vs. KK example, the edge of the AA side shrinks rapidly as you introduce an imperfect shuffle. Of course it does. But still, if I'm able to pick one of the sides, my odds should never get to worse than 50/50. If in doubt, I take a (unbent) coin out of my pocket and flip it. If the other guy chooses a side and I'm forced to take the one he's willing to give me, then that's a different situation, and of course one I would not like my odds in.

A friend of mine and I do this thing where we make a bet on a baseball game every week. One person picks the bet, and the other person chooses a side (and then we swap next week). If he picks something that's hard for me to judge which side has the edge, and/or I think he might be trying to deceive me into the wrong side with something that's deceptively attractive, I can always flip a coin and accept my 50/50 shot. It doesn't matter that I don't "know" whether or not Aaron Harang has a >50% chance of lasting more than 6 innings.

In the coin example there's no way to think I might have any insight at all into which side the coin will fall on. So I accept that, and pick at random. And I have a 50% chance of picking correctly.

[NotReady]

[ QUOTE ]

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


[/ QUOTE ]

I don't see any problem with anything you said except where did you get the 11 to 10? If the bend is extremely slight then you could have a bankroll problem. And if the guy providing the coin knows it's bent and offers 11 to 10 I wouldn't go for less than 5 to 3. Otherwise "you're going to wind up with an ear full of cider".

[wtfsvi]

[ QUOTE ]
I don't see any problem with anything you said except where did you get the 11 to 10? If the bend is extremely slight then you could have a bankroll problem. And if the guy providing the coin knows it's bent and offers 11 to 10 I wouldn't go for less than 5 to 3. Otherwise "you're going to wind up with an ear full of cider".

[/ QUOTE ] This does not make any sense. You're +EV at 11 to 10 if the coin is bent very slightly, very heavily, or not bent at all. It doesn't matter at all for your EV. You have a 50% chance of picking the right side and get paid 11 to 10.

[T50_Omaha8]

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.

[jason1990]

[ QUOTE ]
A coin that has two identical sides, but we don't know if they are heads or tails, is the same as the OP. And in such a scenario, we can assume a 50% chance of heads if we flip the coin.

[/ QUOTE ]
If I meet you in person, I will bring such a coin. I will tell you it has two identical sides, but I will not tell you they are both heads. Since you will assume a 50% chance of heads, you will be happy to accept my wager when I offer you 3 to 2 odds that it will come up tails. It sounds like free money for me.

[ QUOTE ]
A 11/10 bet on the bent coin will be +EV no matter what method we use to pick heads/tails.

[/ QUOTE ]
I have a coin that I bent last year. It has come up heads 992,178 times in the last million flips. I would like to suggest a method for you to use to pick heads/tails. Go buy a lottery ticket. If you win, pick heads. If you lose, pick tails. I will offer you 11 to 10 odds on this bet. Do you think this is +EV for you?

[ QUOTE ]
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.

[/ QUOTE ]
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.

[wtfsvi]

[ QUOTE ]
you will be happy to accept my wager when I offer you 3 to 2 odds that it will come up tails. It sounds like free money for me.

[/ QUOTE ] No. This is were your argument trips. The moment you offer me this I do know something about the probability of the two outcomes. Therefore I will not take your bet.

[NotReady]

[ QUOTE ]

It doesn't matter at all for your EV. You have a 50% chance of picking the right side and get paid 11 to 10.


[/ QUOTE ]

It does matter because you base your future bets on the result of the first flip so you odds of picking the right side will depend on the amount of the bend.

[Phil153]

This whole thread come to down to DS et al saying "if you have only this information" and jason saying "but what if you have other information!"

Jason's only real point seems to be that probability models need to take into account all available evidence, and not just a portion of it. He should just say so because I don't think there's anyone here who disagrees. A far more interesting topic would be whether probability modelling (taking into account ALL the evidence) can be useful in some situations. Or whether people fail to use probability when it would yield useful results.

I think everyone here is intelligent enough to not need a cautionary tale of a two-headed coin. The strange thing is he thinks it's fine to use probability in the AA vs KK example, where you could just as easily posit a card mechanic as a two headed coin. The amusing thing is that he's used straight up probability in both cases anyway to make his point - the odds of being hustled in a coin wager vs the odds of getting a fair deal in a poker game.

[wtfsvi]

[ QUOTE ]

It does matter because you base your future bets on the result of the first flip so you odds of picking the right side will depend on the amount of the bend.

[/ QUOTE ] You don't base your future bets on the results of the first flip in the scenario in the OP.

[jason1990]

[ QUOTE ]
In the coin example, the person who bent the coin does not pick a side.

[/ QUOTE ]
He does. From the OP:

[ QUOTE ]
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.

[/ QUOTE ]
I bent it. I picked the side. The side is heads.

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.

[NotReady]

[ QUOTE ]

You don't base your future bets on the results of the first flip in the scenario in the OP.


[/ QUOTE ]

From the OP:

[ QUOTE ]

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.


[/ QUOTE ]

[vhawk01]

[ QUOTE ]
[ QUOTE ]
A coin that has two identical sides, but we don't know if they are heads or tails, is the same as the OP. And in such a scenario, we can assume a 50% chance of heads if we flip the coin.

[/ QUOTE ]
If I meet you in person, I will bring such a coin. I will tell you it has two identical sides, but I will not tell you they are both heads. Since you will assume a 50% chance of heads, you will be happy to accept my wager when I offer you 3 to 2 odds that it will come up tails. It sounds like free money for me.

[ QUOTE ]
A 11/10 bet on the bent coin will be +EV no matter what method we use to pick heads/tails.

[/ QUOTE ]
I have a coin that I bent last year. It has come up heads 992,178 times in the last million flips. I would like to suggest a method for you to use to pick heads/tails. Go buy a lottery ticket. If you win, pick heads. If you lose, pick tails. I will offer you 11 to 10 odds on this bet. Do you think this is +EV for you?

[ QUOTE ]
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.

[/ QUOTE ]
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.

[/ QUOTE ]
Your suggestion wasn't that you offer him 3:2 on a HEADS bet, it was that you offer him 3:2 and he can pick either heads or tails without knowing which it is. The fact that you know doesn't change unless you are the one picking. I really hope your entire point doesn't come down to this obvious misunderstanding.

[wtfsvi]

[ QUOTE ]
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.

[/ QUOTE ] 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.

[jason1990]

The whole two-headed coin thing is completely tangential to my original point. My original point addresses people, like yourself, who say this:

[ QUOTE ]
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.

[/ QUOTE ]
Here is my original point, rephrased and located elsewhere in this thread:

--------------------

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.

[NotReady]

[ QUOTE ]

I'm not quite sure what your misunderstanding is, so it's hard for me to address, but I'm quite sure you misunderstand.


[/ QUOTE ]

I should save this kind of frustration for my posts on theism - at least then I feel the aggravation has a good purpose.

[jason1990]

[ QUOTE ]
Jason's only real point seems to be that probability models need to take into account all available evidence

[/ QUOTE ]
This is not my point at all. I know my posts are esoteric, but keep reading. You should be able to understand my latest ones.

[wtfsvi]

[ QUOTE ]
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]

[ QUOTE ]
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.

[/ QUOTE ]

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

[jason1990]

[ QUOTE ]
Your suggestion wasn't ...

[/ QUOTE ]
My only suggestion was this:

[ QUOTE ]
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.

[/ QUOTE ]
That is the inspiration for this thread. This might say it all:

[ QUOTE ]
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.

[/ QUOTE ]

[T50_Omaha8]

[ QUOTE ]
This might say it all:

[ QUOTE ]
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.

[/ QUOTE ]

[/ 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]

[ QUOTE ]
[ QUOTE ]
This might say it all:

[ QUOTE ]
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.

[/ QUOTE ]

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

[/ QUOTE ]
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
--------------------------
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.
---------------------------------

The important practical point
-----------------------------

[ QUOTE ]
[ QUOTE ]
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.


[/ QUOTE ]

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.

[/ QUOTE ]


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]

[ QUOTE ]
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.


[/ QUOTE ]

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.

[PairTheBoard]

[ QUOTE ]
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.

[/ QUOTE ]

I could say the same about yours here,

[ QUOTE ]
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.

[/ QUOTE ]

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]

[ QUOTE ]
[ QUOTE ]
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.

[/ QUOTE ]

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?

[/ QUOTE ]

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]

[ QUOTE ]
Probability does not apply to real objects.

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

[ QUOTE ]
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.

[/ QUOTE ]
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]

[ QUOTE ]
[ QUOTE ]
Probability does not apply to real objects.

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

[/ QUOTE ]

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

[Phil153]

[ QUOTE ]
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?

[/ QUOTE ]
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.

[blah_blah]

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]

[ QUOTE ]
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.

[/ QUOTE ]

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]

[ QUOTE ]

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

[/ QUOTE ]

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]

[ QUOTE ]
Here are my answers:

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

[/ QUOTE ]
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.

[Piers]

[ QUOTE ]
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.

[/ QUOTE ]

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]

[ QUOTE ]
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.

[/ QUOTE ]
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]

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

[/ QUOTE ]

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?
[ QUOTE ]
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

[/ QUOTE ]

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

[/ QUOTE ]
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]

[ QUOTE ]
[ QUOTE ]

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

[/ QUOTE ]

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.

[/ QUOTE ]

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]

[ QUOTE ]
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.



[/ QUOTE ]

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]

[ QUOTE ]
[ QUOTE ]
Here are my answers:

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

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

[/ QUOTE ]
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?

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

[/ QUOTE ]
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]

[ 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.

[/ QUOTE ]

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.

[/ QUOTE ]

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]

[ QUOTE ]

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.


[/ QUOTE ]

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.