Reconcile this...ML Can't

Given the count and number of cards left there is a pack composition that most perfectly matches that information, and has a mean that is the mean of all the possible packs with that information applying, but---

---it is also definite that that composition is not representative for the compositions of those packs that actually have such numbers.

Contradiction?

Actually not. In all such cases the compositions are known to be different from the perfectly mean composition, and each departure from that composition is the starting point of all the distributions of compositions that can result. The mean composition covers all, but is known to be off for any given actual pack.

That is why I based my proof on the True Count Theorem, where when a pack is not defined by the most probable composition given only your main count(s) and the cards left, you have to add various side counts, where the true count of each added side count is also subject to the TCT, and thus limits further distribution. You are increasingly pulled away from the mean most perfect compositions, making them: representative for all possible packs given the count and cards left, but not representative for any given initial pack dealt to that number of cards left and having that count.

This can alternatively be illustrated by using Griffin's Unit Normal Loss Integral tables from The Theory of Blackjack. Every starting pack has an expected average departure from any prior mean that is basically a function of the number of cards from the start of the pack to the current penetration in that pack. This gives another model for what I described more simply in my proof Blame Woggy etc. Every such interval of pack penetration gives you a prediction on deviation from such perfect compositions, where the apparent contradiction above, that a composition can be simulaniously representative of all packs given the count and cards left, but not representative of all probable packs given the number of cards initially, and the count, and the cards left.

What ML discusses is a false definition that is so false his errors in this must be deliberate. He omits how, see the chart in TOB on the spectrum of opportunity given different sized initial packs, having X cards left out of 8 decks initially, is different from having X cards left out of 2 decks initially. If you apply the formula from Thorp and Walden's paper on the spectrum of opportunity from counting, (see bjmath.com) the difference in the opportunity is closely related to the difference in basic strategy numbers, where the effects in being pulled from the mean composition(s) exactly cancel any gains from the mean composition(s) resembling initially smaller packs. It is a bit advanced to post on th free pages and the usual suspects would lock out any posting on this on the closed boards.

But the best retort to this is still my proof with perhaps this addition for ML:

As you require new counts to define the actual composition the actual compositions more closely resemble packs relating to the extreme subsets for single denomination counts.

This would be added to the proof section for Postulate IV for my proof. It would show how bow effects are amplified not just by having more extreme true counts overall, but also are amplified for the side counts (as per my proof) being driven into more extremes as well.

But ML's view is that the definition of mean composition, derived from representing all possible packs given count and cards left, is absolute, and leads of course to the absolute absurdity that he keeps posting.

At this point I have to leave the absurd to the absurd. There is no convincing the clones. Alas this is not the first time, or the first area of expertise, where I have been ahead of the curve and smitten by the blows of those whose mental abilities required the curve be instituted in the first place. I simply must refuse to play or post at their level of understanding, or refusal to understand.

Cant keeps calling the expected values which were clearly defined the same way in my posted proof as competent mathematicians define expected values as being the mean expectation of a class of expectations as the samples of the random variables constituting the class approach infinity as a mean pack composition when it is not that at all. Since he apparently does not yet understand the mathematical concept of an expected value even after I have repeated the definition over and over and referred him to references, I guess he will never learn. But he tells a damn lie every time he says the expected values in the proof are not what any person with a slight knowledge of mathematics knows are expected values.

Cant: Given the count and number of cards left there is a pack composition that most perfectly matches that information, and has a mean that is the mean of all the possible packs with that information applying, but---

ML: That might be true but it has nothing to do with the definition of expected value. Read the definition again.

Cant ---it is also definite that that composition is not representative for the compositions of those packs that actually have such numbers.

ML: That is an unsupported assertion which I think is untrue but, if true, is still irrelevant as to the concept of expected values. Since I never used the first mean or "perfect combination" (except as described by the concept of expected value) in the proof and Cant continuing to say I did do that is just a conscious lie or extreme ignorance. Anyone not a mathematician can check with a mathematician to confirm that was the concept I used.

Cant: Contradiction?

ML: Makes no difference. Apples and oranges.

Cant: Actually not. In all such cases the compositions are known to be different from the perfectly mean composition, and each departure from that composition is the starting point of all the distributions of compositions that can result. The mean composition covers all, but is known to be off for any given actual pack.

ML: That does not affect the value of the expected value at all.

Cant: That is why I based my proof on the True Count Theorem, where when a pack is not defined by the most probable composition given only your main count(s) and the cards left, you have to add various side counts, where the true count of each added side count is also subject to the TCT, and thus limits further distribution. You are increasingly pulled away from the mean most perfect compositions, making them: representative for all possible packs given the count and cards left, but not representative for any given initial pack dealt to that number of cards left and having that count.

ML: How many times does it take even a dumbass like Cant to understand expected value by necessity takes those things into account even if possibly a mean distribution does not. I never used the mean distribution, never said I used the mean distribution, would never use the mean distribution in a proof where I was comparing expected values to an undefined value which was what I did and anybody even slightly knowledgeable in math would know I did. There was not a slight bit of ambiguity in the terminology or the definitions. Cant keeps insisting there was but he is flat wrong.

Cant: This can alternatively be illustrated by using Griffin's Unit Normal Loss Integral tables from The Theory of Blackjack. Every starting pack has an expected average departure from any prior mean that is basically a function of the number of cards from the start of the pack to the current penetration in that pack.

ML: Bullshit. The chart Griffin has called the Unit Normal Linear Loss Integral has nothing to do with expectation. It is a chart to determine the efficacy of strategy plays at different levels and shows the efficacy goes up with increased penetration. The proof deals with expected values only for basic strategy. Cant says the chart stands for "Every starting pack has an expected average departure from any prior mean that is basically a function of the number of cards from the start of the pack to the current penetration in that pack." when it means nothing remotely like that. See page 86 of the Elephant edition where the concept is discussed only in terms of play variations, not expectation.

Cant: This gives another model for what I described more simply in my proof Blame Woggy etc. Every such interval of pack penetration gives you a prediction on deviation from such perfect compositions, where the apparent contradiction above, that a composition can be simulaniously representative of all packs given the count and cards left, but not representative of all probable packs given the number of cards initially, and the count, and the cards left.

ML: Double bullshit. It talks about play variations in a chapter on play variations has has nothing to do with expectations. Anyone can see that for themselves.

Cant: What ML discusses is a false definition that is so false his errors in this must be deliberate. He omits how, see the chart in TOB on the spectrum of opportunity given different sized initial packs, having X cards left out of 8 decks initially, is different from having X cards left out of 2 decks initially.

ML: That is crazy. Expected value is expected value. Of course expected values would be different when starting with eight or two decks. In all I have written about blackjack, you cannot find a place I said expectations are different. You had to make this up.

Cant: If you apply the formula from Thorp and Walden's paper on the spectrum of opportunity from counting, (see bjmath.com)

ML: Please do see the paper as proof of how dishonest Cant is. More on that follows.

Cant: the difference in the opportunity is closely related to the difference in basic strategy numbers, where the effects in being pulled from the mean composition(s) exactly cancel any gains from the mean composition(s) resembling initially smaller packs.

ML: The paper says nothing of the sort. It uses the earlier Thorp and Walden paper as a steppingstone to look at some other games not blackjack in relation to the advantage which might be gained by knowing card depletion. Nothing about basic strategy numbers is mentioned. There is nothing about cancellation. It is all about prediction, expected values of gains from depleted decks, if you will.

Cant: It is a bit advanced to post on th free pages and the usual suspects would lock out any posting on this on the closed boards.

ML: But it is posted on bjmath and does not have to be posted here. It may not be too advanced for the free pages but it is certainly too advanced for you. You did not describe a thing about it correctly just as you misrepresented the Griffin reference as dealing with expectation.

Cant: But the best retort to this is still my proof with perhaps this addition for ML:

As you require new counts to define the actual composition the actual compositions more closely resemble packs relating to the extreme subsets for single denomination counts.

Makes no difference. I started out using expected value and still am. I never used a "perfect" or "mean" composition to stand for expected value. I never would.

Cant: This would be added to the proof section for Postulate IV for my proof. It would show how bow effects are amplified not just by having more extreme true counts overall, but also are amplified for the side counts (as per my proof) being driven into more extremes as well.

ML: I do not know why? Nothing that Griffin said or Thorp said has the slightest bit of relation to what you have said here. That is easy for anyone to confirm. All they have to do is look it up.

Cant: But ML's view is that the definition of mean composition, derived from representing all possible packs given count and cards left, is absolute, and leads of course to the absolute absurdity that he keeps posting.

ML: That is not my view, never has been my view, never been described by me as my view, and is totally irrelevant to my proof which used expected values, an entirely different thing. It is so stupid for you to keep saying I was not using expected values when it is so clear to anyone with any knowledge of mathematics I did.

Cant: At this point I have to leave the absurd to the absurd. There is no convincing the clones.

ML: There is no convincing anyone the proof did not deal with expected values of true counts. That was the unambiguous definition contained in the proof and that unambiguous definition squared with the unabmbiguous definition of expected value.

Cant: Alas this is not the first time, or the first area of expertise, where I have been ahead of the curve and smitten by the blows of those whose mental abilities required the curve be instituted in the first place.

ML: Alas, genius who is so far above us that you cannot even accurately describe what is written by Griffin and Thorp.

Cant: I simply must refuse to play or post at their level of understanding, or refusal to understand.

ML: Great idea. Quit posting and you will not show your ass as often.

Too much time on your hands. :-)

Except in the inbred world of the clones, there are no compitent mathematicians who subscribe to any claim whatsoever that sampling a population in any way modifies the mean of that population, or its unsampled remainders.

The remaining pack of cards is a remaining population BTW!

In case ML wishes to still claim his responses are his actual opinions he should note that, very simply, Abdul Jalib himself (rec.net discussions and bjmath.com archives) noted that the true count theorem in fact applied to any count that would end in zero. I don't see how he, unless trying to make sure the clones remain confused on this issue, could argue about my Axiom V, that changes in mean expectation, due to changes in composition, are in proportion to the true counts tracking those composition changes.

So you have to procede to my Postulate IIb conclusion, unless like ML you wish to manipulate definitions of the word mean, or the phrase mean expectation, and add to it ridiculous statements about how one has to accept his formats and flawed postulates to be allowed to challenge his conclusions, and that only those that agree with him are real mathematicians.

Funny how I have a collection of email responses from several university math departments on how ridiculous ML's claims are, in that they amount to a claim that the act of sampling a population changes the overall mean for the unsampled remainder of that population. Include in that responses from the American Society of Statisticians and the American Mathematical Society.

There is no factual content in the Cant post above. You will notice Cant did not mention his misunderstandings of Griffin and Thorp pointed out but instead went off on another of his tangents. Typical Toll stuff.

Cant: Except in the inbred world of the clones, there are no compitent (sic) mathematicians who subscribe to any claim whatsoever that sampling a population in any way modifies the mean of that population, or its unsampled remainders.

ML: And not among the clones so far as I know.

Cant: The remaining pack of cards is a remaining population BTW!

ML: Somewhat deceptive. Any fifty-two, for example, cards which might remain in a real game is only one of the subsets one would use for analysis and there are billions of subsets in very short order because there are m! ways just for stacks of the same cards and depletion allows all kinds of different card combinations to occur. You seem to be suggesting there is some significance to be attached here when the significance is 1/billions. Analysis requires total analysis of all the subsets.

Cant: In case ML wishes to still claim his responses are his actual opinions he should note that, very simply, Abdul Jalib himself (rec.net discussions and bjmath.com archives) noted that the true count theorem in fact applied to any count that would end in zero.

ML: Jalib used expected values in his proof of the theorem, BTW. The same think I used in my proof. But you will be unable to further explain the relevance of that statement to my proof because it has no relevance. Please explain the relevance.

Cant: I don't see how he, unless trying to make sure the clones remain confused on this issue, could argue about my Axiom V, that changes in mean expectation, due to changes in composition, are in proportion to the true counts tracking those composition changes.

ML: Because it is irrelevant to expectected values. Common and odd departures happen and they are all summed to calculate an expected value for a group of random variables.

Cant: So you have to procede to my Postulate IIb conclusion,

No you do not. Your conclusion is not mathematically supported and it really makes no mathematical sense whatsoever.

Cant: unless like ML you wish to manipulate

ML: What is manipulating to define an expected value in identical terms it is described on mathworld

Cant: definitions of the word mean,

ML: Now that is a nosegrower if I ever saw one. The expected value happens to be a mean by definition and beyond manipulation. Of, couse it is apparent the concept of expected value is beyond Cant's intelligence and he continues to describe the expected value as something it is not.

Cant: or the phrase mean expectation,

ML: And the nose got a little bigger. I used the term expected value and further described it as mean expectation which it surely is. All the literature defines expected value that way.

Cant: and add to it ridiculous statements about how one has to accept his formats

ML: Not ridiculous at all. It is how proofs are criticised by mathematicians to show flaws in definitions and steps.

Cant: and flawed postulates

ML: The nose just grew some more. I said in the notes to my proof that if anyone wanted to attack the proof on the basis of the theorems or postulate to have at it.

Cant to be allowed to challenge his conclusions,

ML: Of course. A mathematical proof must be challanged as to its theorems, postulates, and math to attack its conclusions. That is elementary.

Cant: and that only those that agree with him are real mathematicians.

ML: Another six inches. It is just no real mathematician has disagreed with me yet. You have shown your lack of mathematical knowledge.

Cant: Funny how I have a collection of email responses from several university math departments on how ridiculous ML's claims are, in that they amount to a claim that the act of sampling a population changes the overall mean for the unsampled remainder of that population. Include in that responses from the American Society of Statisticians and the American Mathematical Society.

ML: I do not believe you have anything like that but I would write a testimony to the effect the act of sampling does not change the overall mean for the set or the remainder. That is elementary. My definitions in no way amount to a claim different from that. The definition of expected value in no way claims that and I have consistently said the expected value is all inclusive, exact because the expectation of the finite subsets used in my proof would be totally sampled as the sample size approaches infinity.

Don't trip over your nose on your way out of the library where you are taking up space and resources a student could use because, despite your great blackjack ability and great wealth you derive from it, you do not even have a computer, not even an Atari.

Ever hear of privacy?n I keep my computers away from what I post on the blackjack boards!

Only posting from public terminals.