So What's This Problem I Won't Answer?

[David Sklansky]

I wasn't trying to avoid answering it. In fact I just skimmed it originally. And I thought I answered it with my definition of probability which I will repeat in a minute.

But here is the thing. It doesn't matter whether there is some kind of thechnical flaw in my definition. Because I never had any in intention of arguing rigorous points about mathematics. All such arguments do is give people an excuse to be mathematically illiterate. "Why should I learn this stuff if even mathmeticians can't agree about it." "Why should I learn how to integrate new evidence in my probabalistic opinion about something."

My way works. Period. Unless the new evidence is not independent of the old evidence. Perhaps there is an esoteric exception. But if there is, it should probably not even be mentioned to the average person, again because he will use that fact as an excuse. To not realize that he should convict a defendent with otherwise dubious evidence against him if it turns out he has the same extremely rare blood type as the murderer. Or that he should acquit the defendent where there is very strong evidence that she is a witch, merely because it is almost certain there are no witches.

Anyway before someone reasks me the question which I will answer, let me say again what I mean by "probability".

Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information. There is no "right" answer. Is the next card turned going to be the ace of spades. You will give a probability estimate based on what cards you have seen. Even a fair coin is not even money when flipped if you have some physical information about the flipper.

When the information is not as oviously correlated with probability as seen cards, you resort to your knowledge of that information. A black and a white horse are about to race. You know nothing else about them. If you happen to have knowledge of other races involving black and white horses, say the black horse won 45,000 out of 100,000 you should use that ratio.

If you have information beyond the number of choices but no large data samples, (I say large because if it is small your knowledge about small samples comes into play) you have to use subjective judgement about what percentage of the time you think this information results in one outcome or the other.

If the only information you have is the number of choices, your personal probability should be equally divided among them. Perhaps because you have data that shows that when the only information is the number of choices the choices have come up equally. But in this particular case even if you don't have the data it is silly to say that you are falling back on subjective judgement. Rather you are falling back on not being a moron.

Now if you still need to, ask the question again.

[soon2bepro]

What is the question, again?

(PS: all of this should be quite obvious to any professional gambler. Or any thinking individual for that matter)

[bigmonkey]

I don't think anyone is really wrong on this question. The people arguing are both just using equally valid and also largely compatible interpretations of the probability calculus.

Bayesian Epistemologist- P(coin landing heads) = P(h/E), and given that you have no evidence other than that some relevant evidence is available to some other person (but you cannot infer the nature of it by observing their behaviour -i.e. their betting patterns), then you are forced to conclude that P(h/E) = 1/2.

Propensity theorist- There is some real number that exists objectively in the world that represents the degree to which this coin will land heads in the long term. From the little evidence we have we may say the P(heads) is between 0 and 1 but as this is an axiom in the probability calculus anyway it doesn't really count as evidence. We could and should still accept bets using our Bayesian conception of probability. For example we might accept a bet that pays 10:1 where a fair coin is tossed and if it comes heads we have to bet on heads with the bent coin, and if it comes tails we have to bet on tails with the bent coin. You should still make this bet while being a propensity theorist and maintaining that the probability of heads/tails is unknown.


The difference between these interpretations is that we tend to use the former in our accounts of epistemology, rational action and belief revision, and the latter in our accounts of metaphysics. The first is subjetcive, the second is objective, but they are both compatible.

[Piers]

[ QUOTE ]

If the only information you have is the number of choices, your personal probability should be equally divided among them.

[/ QUOTE ]

If all you know is the number of choices, then you can make a default assumption if you want. Guess equally divided makes as much sense as any other choice. The problem is that this assumption carries no weight. As soon as you get any further information, like you find out what the choices are, that new information should be the only thing you use.

Usually when you use information to form a probabilistic estimate, then get some further information the original estimate should not be discarded but incorporated.

Consequently the quoted statement can be considered of academic interest only. Once you define the problem you are analysing you should discard entirely your initial estimate and only use the new information to create your estimates. I can not see how the quoted estimate can be used to help you in any practical situation.

[Phil153]

This whole thing just comes down to the definition of probability. David is using it in a real world, useful sense and Jason is thinking there is a "real" probability hidden somewhere in the muck that we cannot know or even put a range on without all the information. That may be true, but it's hardly a deep or useful insight.

The trouble is that Jason goes on to claim that if we don't have all the information, we're being totally subjective and there is no way to differentiate the objectiveness of probability estimates. This is just plain silly.

I know who I'll be backing in a game of poker.

[PairTheBoard]

[ QUOTE ]
Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information.

[/ QUOTE ]

They can have two different assessments because they use two different methods for evaluting that information. Those differing methods do not necessitate more information. They may differ according to differing assumptions that are made. Their differing conclusions are then based not on different information but on different assumptions. For the conclusions to be meaningful they need to include a statement of how the conclusion depends on the assumptions being used to arrive at it.

The differing assumptions may be debatable. When a Conclusion is reached that fails to state the assumptions on which is was based, it should be considered suspect. When a person proclaims that Conclusion without considering the merits of adopting other assumptions for the evaluation of the information, his intellectual integrity becomes suspect.

PairTheBoard

[KipBond]

[ QUOTE ]
Now if you still need to, ask the question again.

[/ QUOTE ]

I have a practical(*) application question.

I will bend a coin so that it favors one side over the other. The coin will then be flipped 1,000,000 times. If you guess the precise # of times it will come up heads, you win $1,100,000. If you don't, you pay me $1. We will agree ahead of time to do this bet 10,000,000 times using the same coin and your same guess all 10M times. What is your guess (how many times will the coin come up heads)?

If you don't think this is a fair bet, please explain why. I'm giving you 11:10 odds.

(*) This bet can actually be simulated using a computer program in very little time.

[David Sklansky]

The question as to what is my best guess is not easy. But the bet is more than fair as I have the best of it if I choose a random number between one and a million.

[KipBond]

[ QUOTE ]
The question as to what is my best guess is not easy. But the bet is more than fair as I have the best of it if I choose a random number between one and a million.

[/ QUOTE ]

If we were flipping a fair coin, would you still choose a random #? Why or why not?

And, you meant between ZERO & 1 million, right? /images/graemlins/laugh.gif

I figure there is a 99.9999% chance that I will win $10M in this bet.

[PairTheBoard]

[ QUOTE ]
I figure there is a 99.9999% chance that I will win $10M in this bet.

[/ QUOTE ]

That doesn't do you much good EV-wise if you have a 0.0001% chance of losing $11 trillion. Although if you both pay up until you go broke and David can cover the $10 million but you can't then you might have something.

[ QUOTE ]
If we were flipping a fair coin, would you still choose a random #? Why or why not?


[/ QUOTE ]

This is an interesting question. It shows how the 50% best guess for P(heads on first flip) could be misused. Sklansky shows he knows enough not to misuse it in this situation. Can he explain how he knows? He would surely guess 500,000 for the fair coin which he also assigns the personal probability P(heads on first flip of fair coin)=50%. Why doesn't he guess 500,000 after assigning the personal probability P(heads on first flip of bent coin)=50%? oops, he never actually said that did he? or did he? Nonone really knows.

He talked about personal probabilities based on personal information for cards and horses. He would have no trouble making such a personal probability statement in those cases. Will he come right out and say that for information available to him in the Bent Coin example he assigns the information based personal probability,

P(Heads on first flip of Bent Coin) = 50% ?

btw, David. That's been the question you haven't answered.

PairTheBoard

[KipBond]

[ QUOTE ]
[ QUOTE ]
I figure there is a 99.9999% chance that I will win $10M in this bet.

[/ QUOTE ]

That doesn't do you much good EV-wise if you have a 0.0001% chance of losing $11 trillion. Although if you both pay up until you go broke and David can cover the $10 million but you can't then you might have something.

[/ QUOTE ]

Yeah, I know... I'm willing to take that chance. /images/graemlins/laugh.gif I guess I'm not a very smart gambler. /images/graemlins/wink.gif

[David Sklansky]

Yes. That's my personal probability. Its no different then my personal probability that the next card you turn over will be red even though you burned a card. I would have answered that a long time ago if I thought you really cared. Now what?

[PairTheBoard]

[ QUOTE ]
Yes. That's my personal probability. Its no different then my personal probability that the next card you turn over will be red even though you burned a card. I would have answered that a long time ago if I thought you really cared. Now what?

[/ QUOTE ]

You didn't think we cared? You must not have been paying attention to the "Clarification" thread that you started where it was asked emphatically and explicitly 4 different times. We could only conclude you were dodging the question and we couldn't figure out why you would dodge it. Why did you seem to insist on couching it in a formula whereby you randomize your decision to Call heads or tails with the flip of a 2nd Fair Coin? Something that would insure a fair game for yourself even if you knew the bent coin had a 75% Heads bias.

After you were asked 4 times with links to all 4 times shown on the last request you continued to ignore the question. It looked liked you were avoiding the question because you thought it might be a mistake to say yes to it. Which made us wonder if there was something wrong with your concept of probability in this case. You didn't have that thread on ignore yet either. You had been engaged but simply would not answer the question. In fact it looked like you disengaged from the thread precisely because you were uncomfortable with the question.

Thanks. We can now use your confirmation without having to guess at it anymore.


PairTheBoard

[KipBond]

PairTheBoard:
[ QUOTE ]
P(Heads on first flip of Bent Coin) = 50% ?

[/ QUOTE ]

David Sklansky:
[ QUOTE ]
Yes. That's my personal probability.

[/ QUOTE ]

Then, why is it so hard to pick a # in my wager with you? Clearly, if your personal probability is 50%, then you need to pick 500,000:

My Wager:
[ QUOTE ]
I will bend a coin so that it favors one side over the other. The coin will then be flipped 1,000,000 times. If you guess the precise # of times it will come up heads, you win $1,100,000. If you don't, you pay me $1. We will agree ahead of time to do this bet 10,000,000 times using the same coin and your same guess all 10M times. What is your guess (how many times will the coin come up heads)?

[/ QUOTE ]

David Sklansky
[ QUOTE ]
The question as to what is my best guess is not easy.

[/ QUOTE ]

It should be easy if you are confident in your personal probability assessment.

How about a wager where you are allowed a range of guesses? OK, let's do that:

I will bend a coin so that it favors one side over the other. The coin will then be flipped 1,000,000 times. You select a (contiguous) range of 50,000 #s. If the # of heads results is in your range, you win $21. If you don't, you pay me $1. We will agree ahead of time to do this bet 10,000,000 times using the same coin and your same range all 10M times. What is your guess: what range do you select?

Surely with a fair coin this is extremely simple. The P(H)=50%, so you pick the range: [475,000 - 525,000]. You expect to win a bunch of money over the 10M trials.

If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager -- the one that is guaranteeing you many millions of dollars? Why aren't you using your probability to make the best decision? Why, instead, are you insisting on randomizing your decision?

[luckyme]

[ QUOTE ]
How about a wager where you are allowed a range of guesses? OK, let's do that:

I will bend a coin so that it favors one side over the other. The coin will then be flipped 1,000,000 times. You select a (contiguous) range of 50,000 #s. If the # of heads results is in your range, you win $21. If you don't, you pay me $1. We will agree ahead of time to do this bet 10,000,000 times using the same coin and your same range all 10M times. What is your guess: what range do you select?

Surely with a fair coin this is extremely simple. The P(H)=50%, so you pick the range: [475,000 - 525,000]. You expect to win a bunch of money over the 10M trials.

If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager -- the one that is guaranteeing you many millions of dollars? Why aren't you using your probability to make the best decision? Why, instead, are you insisting on randomizing your decision?

[/ QUOTE ]

This is the issue I'm struggling with. It seems to me the 50-50 used in the original bet is not a comment on the nature of the coin, it's a betting tactic, so it's doesn't translate into a different setup directly.

In your new scenario - I would randomly pick the H or T side on a 50-50 basis, then I'd randomly select a 50K range within that side. That should put me in the winning range 5% of the time, but it still makes no comment on the actual nature of the coin, it protects me from misguessing a rock,paper,scissors move by you.

I'm a Sklamorian, but the above would have me taking the bet for a non-painful amount of money.

ok, straighten me out, luckyme

[jason1990]

[ QUOTE ]
This is the issue I'm struggling with. It seems to me the 50-50 used in the original bet is not a comment on the nature of the coin, it's a betting tactic, so it's doesn't translate into a different setup directly.

[/ QUOTE ]
You are confusing two things. On the one hand, there is David's personal probability (DPP):

[ QUOTE ]
P(Heads on first flip of Bent Coin) = 50% ? Yes. That's my personal probability.

[/ QUOTE ]
On the other hand, there is David's tactic (DT):

[ QUOTE ]
if you flip a "fair" coin, use the the result of that flip to choose a side for the bent coin ... You would win getting eleven to ten.

[/ QUOTE ]
These two things have nothing to do with one another. Here is a proof that DT works:

Let p be the unknown probability of heads in the bent coin. Half the time, David will bet on heads and have probability p of winning. Half the time, David will bet on tails and have probability 1 - p of winning. So overall, his probability of winning is (1/2)*p + (1/2)*(1 - p) = 1/2. End of proof.

Notice that the proof does not use DPP at all. Should we decide to use DT? Of course! The proof is right in front of us. Did DPP help us make this decision? Not at all! It was totally irrelevant and had nothing to do with it.

The point here is this. If David wants to say his personal probability is 50%, then fine. Everyone is entitled to their personal probability. But he has not yet constructed a situation in which he can actually use DPP to either make money or avoid losing money. He thinks he is using DPP when he uses DT. But he is not. That is his confusion, and he has passed that confusion onto many people here.

[TomCowley]

DPP gives him a potential edge against people with less or the same amount of information. DT defends against people with more information. Let's say you're watching a replay of a UFC with two friends, one who has seen it and one who hasn't. Let's say DPP is 50% based on the fight matchup. If the friend who hasn't seen it offers you 2:1 on one side and 1:2 on the other side, you will take 2:1 with the expectation you are getting an edge. If the friend who has seen the fight offers the same bets, you cannot auto-take the 2:1 side, because he'll always offer 2:1 on the losing fighter and you'll always lose. The only way to come out even against somebody with more information (a better estimate) is to randomize which side you take.

[PLOlover]

[ QUOTE ]
Let p be the unknown probability of heads in the bent coin. Half the time, David will bet on heads and have probability p of winning. Half the time, David will bet on tails and have probability 1 - p of winning. So overall, his probability of winning is (1/2)*p + (1/2)*(1 - p) = 1/2. End of proof.

Notice that the proof does not use DPP at all. Should we decide to use DT? Of course! The proof is right in front of us. Did DPP help us make this decision? Not at all! It was totally irrelevant and had nothing to do with it.

The point here is this. If David wants to say his personal probability is 50%, then fine. Everyone is entitled to their personal probability. But he has not yet constructed a situation in which he can actually use DPP to either make money or avoid losing money. He thinks he is using DPP when he uses DT. But he is not. That is his confusion, and he has passed that confusion onto many people here.

[/ QUOTE ]

It might be a misunderstanding/miscommunication/coincidence,
but
[ QUOTE ]
. Half the time, David will bet on heads and have probability p of winning. Half the time, David will bet on tails

[/ QUOTE ]
corresponds to DPP.

[jason1990]

[ QUOTE ]
It might be a misunderstanding/miscommunication/coincidence,
but
[ QUOTE ]
. Half the time, David will bet on heads and have probability p of winning. Half the time, David will bet on tails

[/ QUOTE ]
corresponds to DPP.

[/ QUOTE ]
Actually, it corresponds to the fact that he is using a fair coin in DT.

Half the time the fair coin will come up heads, so David will bet on heads and have probability p of winning. Half the time the fair coin will come up tails, so David will bet on tails and have probability 1 - p of winning.

We used: P(fair coin heads) = 50%

We did not use: P(bent coin heads) = 50%

[PairTheBoard]

[ QUOTE ]
KipBond -
Surely with a fair coin this is extremely simple. The P(H)=50%, so you pick the range: [475,000 - 525,000]. You expect to win a bunch of money over the 10M trials.

If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager -- the one that is guaranteeing you many millions of dollars? Why aren't you using your probability to make the best decision? Why, instead, are you insisting on randomizing your decision?


[/ QUOTE ]

The thing to realize here is that Sklansky's Baysian definition of Probability is not the same as the Frequentist. The Frequentist is the one on which mathematical models of probability are based. That doesn't mean that mathematical probability models can't be applied by Baysians. It does mean we have to treat the two definitions differently. It is not a fair criticism of a Baysian statement of probability to treat it as if a Frequentist had make it.

On David's Thread in the Probablity Forum, jason1990 just provided a link to an explanation by Persi Diaconis of a Baysian definition of probilility. I wonder if Sklansky disagrees with any of it. Basically I think Persi provides some of the extra details and rigor to the definition that illuminates what Sklansky defines as Probability. This is the material we should be looking at to see if Sklansky stays consistent in how he makes use of Probability according to his definition. There's no point in making unfair attacks on his definition.

Persi Diaconis on Subjective Probability (http://www-stat.stanford.edu/~cgates/PERSI/courses/stat_121/lectures/subjectiveprob/)


So when you say,

[ QUOTE ]
If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager

[/ QUOTE ]

you are making an unfair attack. Notice Sklansky did not agree to the proposition,

"P(Heads of Bent Coin)=50% (Not a Sklansky statement)"

He agreed to the proposition,

"P(Heads on first flip of Bent Coin)=50% (Sklansky statement)"

Notice I made the conditions, (Not a Sklansky Statement) and (Sklansky Statement) part of the propositions. You might say you were just lazy about writing the "on the first flip" part because that really doesn't matter to you. But it matters to the Baysian and it matters to Sklansky's definition of probability. To make this perfectly clear, Sklansky did NOT agree to the following proposition,

"P(head on first flip of Bent Coin)= 50% (Frequentist statement)"

Sklansky did Not agree to that proposition. So it's not fair to criticize the proposition he agreed to by treating it as if it were a proposition he did not agree to.

If you read the Persi Diaconis link you will see that the requirement he puts on the Baysian Definition of probability is that it not produce any "Dutch Books". He cannot agree to two probability statements whereby his indifference to the odds they would imply for gambling propositions would allow you to make a Dutch Book on him and win money betting both sides of both propositions and automatically win money because of their inconsistency. For example, Sklansky could not say,

"The 6 sides of the Mystery loaded Die are all equally likely on the First roll"

and then also say,

"P(Mystery loaded die comes up 3 on First Roll) = 1/8".

He cannot agree to both those statements because a gambler could make a Dutch Book on him knowing nothing about how the die is loaded.


So when David tries to determine his answer to your gambling proposition with respect to his assertion,

"P(Heads on first flip of Bent Coin)=50% (Sklansky statement)"

he should be doing so in such a way that a Dutch Book cannot be made on him between his statement above and what he says about your gambling proposition. If you think he is failing to do that you have an argument.


But remember, the only other gambling proposition he has said his assertion implies is that he is indifferent to betting heads or tails on an even money first flip of the bent coin. Being indifferent the only way he sees to decide which to bet on IF FORCED, is to flip a fair coin. Something I do myself sometimes in undecidable situations - although I often second guess the flip.

What that implies about gambling propositions that involve flipping the coin more than once is unclear. It may imply nothing, which makes us wonder about its usefulness. But then again that might be ok. The Usefulness comes from the Procedure that went into making that Baysian probability statement. I think we can interrogate him on whether he is using that Procedure consistently when he decides what to say about your new gambling proposition.

We would also like to know more about exactly what that Procedure is. How easy is it to apply? If we apply it once how can we use the result? Or is the result an isolated thing which we can't build on. Do we have to start all over again each time we get a new problem? Is there any way of incorporating the results of one application of the Procedure into the Procedure we use in the next application?

Your new Gambling Proposition is shedding some light on those questions. You are pointing out how the Frequentist assertion,

"P(head on first flip of Bent Coin)= 50% (Frequentist statement)"

would be easy to build on and apply to your new Gambling Proposition. While Sklansky's assertion looks to be one that can't be built on and applied. He must start from scratch so to speak and apply the Procedure that went into his assertion all over again to respond to your new Gambling Proposition. However, this is not a fair criticism either in this case because the Frequentist assertion is almost certainly False. We don't have that assertion from the Frequentist here. The statement we do have from the Frequentist is,

"P(head on first flip of Bent Coin)= not enough information (Frequentist statement)"

We hardly have a case that we can build much on that statement either. At least Sklansky's statement provides a way for us to gamble on the first flip of the Bent Coin.

PairTheBoard

[PLOlover]

[ QUOTE ]
Actually, it corresponds to the fact that he is using a fair coin in DT.

Half the time the fair coin will come up heads, so David will bet on heads and have probability p of winning. Half the time the fair coin will come up tails, so David will bet on tails and have probability 1 - p of winning.

We used: P(fair coin heads) = 50%

We did not use: P(bent coin heads) = 50%

[/ QUOTE ]

well I was under the impression that the only use of the fair coin was to use it as a 50% frequency producing device. What I mean is that DS says, hey, I will just bet heads half the time and tails half the time. Ok, says DS, how do I do that? Oh, I've got it, DS says, I'll flip a fair coin.

so I think we use P(X)=DPP=50% , where you just have to find something that will give you a DS Personal Probablilty of 50%. The actual figure of .5 comes from there being 1 choice of two states (heads,tails).

So the DPP is just #choices/#possiblechoices, which is 1/(1+1) = .5
So DPP has nothing whatsoever to do with the bent coin.
so probably it was a misspeak or misunderstanding to think that DPP had to do with the bent coin.

[ 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. You would win getting eleven to ten

[/ QUOTE ]
So clearly he is stating plainly that he is not using anything to do with the bent coin, only using the number of choices.

[PLOlover]

[ QUOTE ]
But remember, the only other gambling proposition he has said his assertion implies is that he is indifferent to betting heads or tails on an even money first flip of the bent coin. Being indifferent the only way he sees to decide which to bet on IF FORCED, is to flip a fair coin. Something I do myself sometimes in undecidable situations - although I often second guess the flip.

What that implies about gambling propositions that involve flipping the coin more than once is unclear. It may imply nothing, which makes us wonder about its usefulness. But then again that might be ok. The Usefulness comes from the Procedure that went into making that Baysian probability statement. I think we can interrogate him on whether he is using that Procedure consistently when he decides what to say about your new gambling proposition.

We would also like to know more about exactly what that Procedure is. How easy is it to apply? If we apply it once how can we use the result? Or is the result an isolated thing which we can't build on. Do we have to start all over again each time we get a new problem? Is there any way of incorporating the results of one application of the Procedure into the Procedure we use in the next application?

...
We hardly have a case that we can build much on that statement either. At least Sklansky's statement provides a way for us to gamble on the first flip of the Bent Coin.

[/ QUOTE ]

to be fair the sklansky bent coin thread wasn't about betting on the first flip or one flip, it was more like I'll make a 100 predictions for your next 100 flips.
[ 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. You would win getting eleven to ten

[/ QUOTE ]

so if you have a double headed coin then you have 100 heads, and the guesses should be 50 heads 50 tails, so he breaks even at even money, and wins money if getting better than even money.
So if he breakseven/makesmoney on 100 or 1000 flips, hes gonna be EV0 or +EV for any one flip. I mean, if he's getting 11:10 it's a +EV bet for any single flip, right?

also the initial bent coin scenario doesn't presuppose the bent coin will always be bent the same way I don't think. I mean if you kinda abstract the coin a bit you can see where a bent coin could be "floppy", where it could favor heads for a while, then favor tails for while, like a wax coin that deforms with each flip or something. In that case no frequency tracking/guessing will help you and may in fact hurt you.

[jason1990]

David's argument is like a tricky optical illusion. I think it even confused him. (Either that, or he was being intentionally deceptive, which I do not believe.) Let me take one more stab at trying to help you see this.

I have a fair die, 5 sides are blue, 1 side is red. I will roll it. There are only two alternatives, red or blue. What is the probability of red?

Before you answer that, let me tell you this:

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

Do you believe me? You should, because it is true. If you do not believe me, then you will find many people on this forum, even David himself, who will wager against you. So believe it.

Okay, then. Does this logically imply anything about the probability of red? (Hint: No.)

If you still cannot see this, then I am afraid there is nothing more I can do for you.

[PLOlover]

[ QUOTE ]
David's argument is like a tricky optical illusion. I think it even confused him. (Either that, or he was being intentionally deceptive, which I do not believe.) Let me take one more stab at trying to help you see this.

I have a fair die, 5 sides are blue, 1 side is red. I will roll it. There are only two alternatives, red or blue. What is the probability of red?

Before you answer that, let me tell you this:

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

Do you believe me? You should, because it is true. If you do not believe me, then you will find many people on this forum, even David himself, who will wager against you. So believe it.

Okay, then. Does this logically imply anything about the probability of red? (Hint: No.)

If you still cannot see this, then I am afraid there is nothing more I can do for you.

[/ QUOTE ]

ok, well, there are two distinct questions here.

1) what is probability of red.
2) how to minmax our wagering.

1) is irrelevant to point 2)

ok, so if you bet with 50% frequency you will break even against a 5/6 bent coin. ok.

my only point is that DS never uses point 1) , other than to say it could be anything. It could be 50% as in a fair coin, it could be 5/6 as in your dice example. It's irrelevant. As far as I can see the two sides of the argument boil down to

a) how "unfair" or bent the coin is matters
b) it doesn't matter what the heads/tails ratio is.

It's my understanding that you are on side a), and DS is on side b).

Am I incorrect in any of my thinking above?

what is the deceptivity or illusion? you seem to agree that you would break even, and that the frequency of red or heads or whatever doesn't matter. So what's the illusion?

[PairTheBoard]

I think you're missing the point. The point is that Sklansky's statement on the original Bent Coin OP was illogical. It is also misleading, and in yet another sense it is just plain empty air. Here is his statement from that thread,

Link to Original Bent Coin Thread (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=10628726&an=0&page=3#Pos t10628726)
===================
Sklansky -
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.
==================



The statement is illogical. Look closely at what it says. It translates to this:

Tranlation of Sklansky's Statement
----------------------------
1. I tell Jason A and B are mutually disjoint exhastive events.

2. Jason says you cannot tell from that the P(A). He says you do not have enough information.

3. I say you do know only the information that the two events are mutually exclusive and exhastive. ie. you know only that there are two alternatives.

4. Therefore, this allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose A or B for your bet and get even money on your bets. You would win getting eleven to ten on each of your bets.

5. Thus I have given a valid definition of the probability I criticized Jason for begging a lack of information on. ie. The probability of A. In real life you must use my definintion for P(A) instead of jason's lack of information because you never have enough information.
------------------------------


I think if you make a fair comparison of Skansky's comment and my translation you will see it is a fair translation. It's certainly illogical if events A,B are replaced by ones like I give in my "Rejection of Sklansky" thread or the Red Blue Dice events that Jason gave. The second part of #4, "You would win getting eleven to ten on each of your bets" is purely logically irrelevant. All it says is that you would win if someone gave you a Dutch Book. You don't need any premises or any information about anything to know that. If you look closely, the first part of #4 is also like a Dutch Book. It is a Neutral Dutch Book. So #4 could be even more simply translated to,

4. Give me a Neutral Dutch book and if forced to bet I will take both sides and break even. Give me a Dutch Book and I will take both sides and make money.

So, #4 is basically like saying 1+1=2. It's true but we already know it. It's irrelevant. It illuminates nothing about P(A).


If you criticize replacing A,B with actual Events because those events do not "say only" that A and B are mutually exclusive exhastive events, then you have to observe that the Bent coin example does not "say only" that Heads and Tails are mutually exclusive exahastive events. In fact, any events that are actually described to us will not "say only" that they are mutually excusive and exhastive.

So Sklansky is Misleading people with that statement. He is encouraging them to conclude that sparse information equates to "saying only" that they are mutually exclusive exhastive events. Either that or his statement is simply Empty Air because it never applies with certainty to any mutually exclusive exhastive events that are actually presented to us.

In this Thread Sklansky has clarified his defintion of probability - with a little help by us applying Persi Diaconis' more explicit explanation of subjective probability. With that clarification Sklansky might have said this instead of the above,

Revision of Sklanky's Statement
---------------------------
Jason won't give the Probability of Heads for the Bent Coin begging lack of information. I can give a probability based on my definition of probability, which is an information based probability personal to the information I have available. The information I have here is that there are two alternatives and by their description I see no reason to choose one over the other. So I say P(Heads on First flip)=50%.

What I mean by that is that I am indifferent to a bet on Heads or Tails at even money. By that I mean that if Forced to choose, having no other way to decide seeing what I do in the information contained in the description of the two alternatives , I would flip a coin to decide. However I would not be indifferent if I was offered a bet with better than even money odds.
---------------------------------



At this point it's getting hazy again. What would he mean by that last statement? Does he have to be given better than even money odds on both sides of the bet to not be indifferent to the bet? What if he is only given odds on one side of the bet. Would he not be indifferent to that bet? If offered 1 million to 999,999 odds to take heads with no option to make a bet on Tails, would he make a bet on heads?

I asked that question on one his Threads for this topic but he did not respond. So I'm afraid his definition of information based personal probability is still not completely clear. If he insists on better than even odds on both sides of the bet to not be indifferent to the bet, then he is just insisting that someone gives him a Dutch Book. You need no information about anything to make money on a Dutch Book. You don't have to flip any coins either. You just take both sides of the bet. Insisting on a Dutch book is like saying 1+1=2. It's true but it's something we know already. It illuminates nothing.

PairTheBoard

[PLOlover]

[ QUOTE ]
I think you're missing the point. The point is that Sklansky's statement on the original Bent Coin OP was illogical. It is also misleading, and in yet another sense it is just plain empty air. Here is his statement from that thread,

Link to Original Bent Coin Thread
===================
Sklansky -
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.
==================



The statement is illogical. Look closely at what it says. It translates to this:

Tranlation of Sklansky's Statement
----------------------------
1. I tell Jason A and B are mutually disjoint exhastive events.

2. Jason says you cannot tell from that the P(A). He says you do not have enough information.

3. I say you do know only the information that the two events are mutually exclusive and exhastive. ie. you know only that there are two alternatives.

4. Therefore, this allows you to break even on your bets if you flip a "fair" coin, use the the result of that flip to choose A or B for your bet and get even money on your bets. You would win getting eleven to ten on each of your bets.

5. Thus I have given a valid definition of the probability I criticized Jason for begging a lack of information on. ie. The probability of A. In real life you must use my definintion for P(A) instead of jason's lack of information because you never have enough information.
------------------------------


I think if you make a fair comparison of Skansky's comment and my translation you will see it is a fair translation. It's certainly illogical if events A,B are replaced by ones like I give in my "Rejection of Sklansky" thread or the Red Blue Dice events that Jason gave. The second part of #4, "You would win getting eleven to ten on each of your bets" is purely logically irrelevant. All it says is that you would win if someone gave you a Dutch Book. You don't need any premises or any information about anything to know that. If you look closely, the first part of #4 is also like a Dutch Book. It is a Neutral Dutch Book. So #4 could be even more simply translated to,

4. Give me a Neutral Dutch book and if forced to bet I will take both sides and break even. Give me a Dutch Book and I will take both sides and make money.

So, #4 is basically like saying 1+1=2. It's true but we already know it. It's irrelevant. It illuminates nothing about P(A).


If you criticize replacing A,B with actual Events because those events do not "say only" that A and B are mutually exclusive exhastive events, then you have to observe that the Bent coin example does not "say only" that Heads and Tails are mutually exclusive exahastive events. In fact, any events that are actually described to us will not "say only" that they are mutually excusive and exhastive.

So Sklansky is Misleading people with that statement. He is encouraging them to conclude that sparse information equates to "saying only" that they are mutually exclusive exhastive events. Either that or his statement is simply Empty Air because it never applies with certainty to any mutually exclusive exhastive events that are actually presented to us.

In this Thread Sklansky has clarified his defintion of probability - with a little help by us applying Persi Diaconis' more explicit explanation of subjective probability. With that clarification Sklansky might have said this instead of the above,

Revision of Sklanky's Statement
---------------------------
Jason won't give the Probability of Heads for the Bent Coin begging lack of information. I can give a probability based on my definition of probability, which is an information based probability personal to the information I have available. The information I have here is that there are two alternatives and by their description I see no reason to choose one over the other. So I say P(Heads on First flip)=50%.

What I mean by that is that I am indifferent to a bet on Heads or Tails at even money. By that I mean that if Forced to choose, having no other way to decide seeing what I do in the information contained in the description of the two alternatives , I would flip a coin to decide. However I would not be indifferent if I was offered a bet with better than even money odds.
---------------------------------



At this point it's getting hazy again. What would he mean by that last statement? Does he have to be given better than even money odds on both sides of the bet to not be indifferent to the bet? What if he is only given odds on one side of the bet. Would he not be indifferent to that bet? If offered 1 million to 999,999 odds to take heads with no option to make a bet on Tails, would he make a bet on heads?

I asked that question on one his Threads for this topic but he did not respond. So I'm afraid his definition of information based personal probability is still not completely clear. If he insists on better than even odds on both sides of the bet to not be indifferent to the bet, then he is just insisting that someone gives him a Dutch Book. You need no information about anything to make money on a Dutch Book. You don't have to flip any coins either. You just take both sides of the bet. Insisting on a Dutch book is like saying 1+1=2. It's true but it's something we know already. It illuminates nothing.

PairTheBoard

[/ QUOTE ]

oh i see now.

well DS is basically saying that well you have a rigged coin so bets against you should be -EV, but hey look I can dutch book it at 0EV and not only that but since you have a rigged coin you might even give me odds and guess what in that case my bets are +EV.
cool counterexample yes but not sure about it in an argumentative sense.

now when he goes on to talk about defining probablility and how there is no fair coin and such,
i think he gets into trouble there because like you said his counterexample had nothing to do with the coin, it was a dutch book that was independent of the heads/tails probability.
So limited information about the coin had nothing to do with it, but he implies there that it does.

So i see your point and i think that in the argument you clearly win against DS,
however i think his counterexample is very clever and can be helpful to think about it.

Anyway, at the end of the day this subject helped me learn some stuff about probability so thanks.

[PairTheBoard]

[ QUOTE ]
however i think his counterexample is very clever and can be helpful to think about it.


[/ QUOTE ]

What counterexample? jason's. If so, I agree. jason's was very good and helpful to think about.

PairTheBoard

[PLOlover]

[ QUOTE ]
What counterexample? jason's. If so, I agree. jason's was very good and helpful to think about.

[/ QUOTE ]

the bent coin dutch book. what example was jason's?