Another misrepresentation
Your ignorance of the form and terminology of mathematical proofs or just sheer mendacity makes you continue to misrepresent the terminology. Witness:
Cant: Every true count and penetration combination has associated with them a mean composition, and a mean expectation for that composition.
ML: Absolutely deceptive. It is true every true count and penetration has associated with them (sic) a mean composition but it is mathematical error to say there is a mean expectation for that composition. It would have been accurate if you had said "an expectation for that composition. There is only one sample, the mean composition so there is no mean value, but one value. There you go switching means.
Cant: ML simply demands that one has to use that mean and that mean alone for discussions about the mean for a given true count.
Totally incorrect. The proof used the terminology of expected value of the expectation of subsets of specific true counts at specific depths from an original specific set of cards greater in number than the cards contained in the subsets, not the expectation of the mean composition of the subsets. IMHO the values would be identical at least to all significant figures, but that is not relevant to the fact you continue to say I did not use expected values when the proof clearly demonstrates that is what was used. Saying otherwise is a misrepresentation.
Cant: There are no such requirements.
ML: Agreed. That is why I used expected value rather than the value you say I used and as pointed out above that is not a mean value but instead a set, definite value.
Cant: The use of such mean compositions is simply a useful practice for reducing computations needed to find the mean predicted by a given true count.
ML: Agreed. But that is irrelevant when expected values are being used.
Cant: It is not a requirement to use this mean and never has been.
ML: And I did not do that.
Cant: The actual bow effect is simply the result of the totally predictable difference between the mean expected value for the mean composition associated with a given true count and penetration, and the actual compositions and their expected values.
ML: Disagree but that has nothing to do with the matters being discussed. Again there is not a "mean" expected value of the mean composition but one value. Further, it was to take into account all possibilities and not just the value of the mean composition I used the terminology E(i,m) which included all compositions which might occur and as you say, if there is a bow effect, all the different compositions have to be taken into account. So in your saying the bow effect occurs when all compositions are taken into account as E(i,n) unambiguously does, I take that as an endorsement of my Postulate which stated precisely that point.
Cant: There in fact is no reversion to such means either. As I detailed in my Blame Woggy.... proof, each composition becomes the starting point for all further distributions.
ML: True, there is no reversion to means. That was my reason for using expected values rather than the value of the mean composition in my proof.
Cant: What results is ML adding to this absurd definition requirement
ML: Cant is now saying using expected values is an absurd thing to use in a mathematical proof. Perhaps he is calling Dr. Thorp an absurd person because the proofs in the paper Cant cited uses expected values over and over. I can think of no proofs dealing with gambling type probability topics which do not use expected value. Perhaps Cant would point one out (he won't because he is lying here).
Cant: an equally absurd demand that this is the only definition for real mathematicians to use and that one is not allowed to post any response unless one uses this same definition.
ML: No. I say you cannot attack a mathematical proof by saying an expected value expressed in the proof is not an expected value at all. A proof can only be talked about in the terms of the proof. The propounder of the proof can define. I expressly used expected values and I have the right as the propounder to do that. If Cant had said the Postulate or the Theorems did not apply to expected values, that would be a valid disagreement. Saying the terms were not expected values but, instead, something else is not a valid disagreement. They cannot be something else because I did not define them as something else and unambiguously defined them as expected values.
Cant: Again: there are no such requirements!
ML: There most definitely are. Once the values were defined as expected values, they are expected values.
Cant: The ruler used is not the thing being measured.
ML: For the life of me I cannot see how that statement relates to anything.
Cant: They have different means associated with them.
ML: Expected values don't and nothing except expected values were used. One of the things you are talking about does not even have a mean but has, instead, a value.
Cant: But once you go down this absurd trail with ML,
ML: The absurd trail of using expected values in a proof blazed by Thorp and often trod by Griffin?
Cant: you have to replace a simple difference in measures
ML: What did I replace? I used expected values of expectation throughout
Cant: with the further absurdity that the basic strategy edge, rather than the edge associated with the actual compositions, floats instead.
ML: That statement is absolutely undecypherable. There are many reasons to believe, including the proof, the expected values of expectations for specific true counts of subsets of m size are larger than the expectations for the same true counts of sets of n size which leads one to believe the E(i,m) is larger than E(i,n) when n>m. One reason to believe that is the expectation (not expected value) of one deck before any cards are dealt is larger than the expectation of a stack of eight decks before any cards are dealt. That is comparing a "perfect" set with a median subset of the set. One can think about creating a "nonperfect" -1 eight deck stack by replacing any four large cards with any four small cards and looking at the expectation and then replacing an eight with a small card in the one deck stack and one can say with some certainty the expectation for the 416 cards would be smaller than the expectation for the 52 cards. However replacement is effected in each stack to preserve a -1 count in each stack, I would say it is reasonable to conclude the expectation for the eight deck stack would be smaller than the expectation for the one deck stack because the one deck stack has a .5+ running start. And it is reasonable to conclude the same for +10 or whatever. That the expectation for the same count for the 52 card stack remains higher, however the composition which established the same count seems more than reasonable and that is consistent with the floating edge proposition. The 52 deck stacks have a .5+ head start which ain't shabby.
Cant: And that is a simple replay of an ultimate absurdity: that somehow measurement of a population changes the mean expected value of any remainder of that population.
ML: It is absurd of Cant to make a statement I ever said such a thing. He will be unable to respond and point out a place where I said such a thing. He will not respond to this request because I certainly never have said such a thing.
Cant: That is exactly what is involved when you claim that penetration alone changes expected values,
ML: Penetration does. If it did not the floating advantage would not exist yet it does.
Cant: which is what you are saying when you say that basic strategy edge changes with penetration!
ML: And, there in a nutshell is your ignorance exposed to the world. The statement the expectation for identical true counts change with penetration does not say in any way overall expectation changes for a specific number of cards drawn from a certain larger stack. Saying fifty-two cards drawn from a 104 card stack which, by coincidence, has the same true count as the 104 card stack have a higher expectation than the original stack is not the least bit inconsistent with saying all the possible 52 card subsets which can be drawn from the 104 card stack have the same expectation as the original 104 card stack. In fact, Cant not recognizing both might very well be true demonstrates how really ignorant Cant is of mathematics and about the fundamentals of blackjack mathematics while posing as conversant with the works of the masters.
The reason both might well be true and there are no inconsistencies is a very simple concept. Any stack of cards has a value for expectation and can be divided into subsets having an average expected equal to the set. The subsets have true counts which spread and if you take the expected value of the expectation of all the true counts involved and average them, some (larger than the starting true count for sure) will have expected values larger than the original expectation and some will have expected values of expectation smaller. What the individual magnitude of each might be is not important so long as the average expected value of all add up to the starting value. Cant has talked about imposing restrictions but it is imposing a restriction to say the expected values of the samples with a true count identical to the true count of the starting set have to equal the value of the starting expectation. The expected values may be larger if the average expected values of all the other true counts are smaller, i.e., steps 4 and 5 of the proof.
And my proof adequately explains how such a thing can happen.
Cant cant be trusted when he talks math. He knows no math.