Proof of the Strong and weak means

Proof of the weak and strong means:

The strong version of the floating advantage alleges that for every mean deck and true count the expectation will have a new mean at every level of penetration matching that of smaller initial full packs.

A precise mean composition pack only occurs at every � deck of penetration only if one each card rank, ace-9, and 4 10s are removed for blackjack. Any other removal results in a density that is only possible from starting with the full pack.

Even if a pack reaches such a perfect mean point at some � deck level the composition is still altered by being part of a larger than that � deck level perfect mean if a round of play crossed that � deck level. The rules of the game still restricted the cards drawn across that barrier.

Thus it is exceedingly rare for any pack to deplete past any � deck barrier in any way that its composition range is that of a perfect mean composition. It is several orders of magnitude rarer for a pack having a composition impossible for a mean pack at � deck in, to be depleted to a perfect mean composition � deck in.

This can be proved from the true count theorem which states: when a pack has a given true count, the means of all subsequent true counts will be that same true count. The same applies to any alterations of means, and to any true count. The mean composition always shifts with penetration, yet every observed composition is the mean composition of all subsequent sub-decks and compositions. Each shift from any perfect mean further constrains the probability of any perfect mean composition.

Thus, rather than centering on the perfect mean at each penetration level, expectation has the same mean as the full pack excluding only the exceedingly rarer, as penetration increases, probability that the pack has crossed a � deck level without being bridged by a round, where that perfect mean occurred. Even then it is still rarer for the mean not to lag behind the current penetration. An 8 deck shoe that has hit a 4 deck perfect mean is far more likely to be constrained to some 4 deck subset mean than to be at a 1 deck mean at 1 deck.

The Bow-effect derives equally well from 2 propositions:
(1) Pushes limit the rise in expectation for the true count in extreme high ranges, and dealer�s hitting strategy is more effective than basic strategy in extreme low ranges. Thus the true count overestimates player edge in all extreme ranges and, in that the mean is still the same as the overall basic strategy edge, underestimates the edge in middle ranges. When the true count is more extreme, with deeper penetration, more of the true count is distributed in such extreme ranges and the bow-effect is increased.

(2) The increasing deviation from the assumed mean decks for each true count, as penetration increases, tends to favor the player in middle true count ranges and the dealer in more extreme ranges.

Either way the bow-effect is derived, it does not involve any floating advantage. Altering your betting ramp to try to match the bow-effect is a zero sum proposition in that your highest spread bet has its upper range cut to the same degree that the middle bets have more edge.

Increases in playing gains break the balance that results in the bow-effect by replacing basic strategy with linear gains (they stay linear longer than the betting prediction with basic strategy) cutting the drop-off in the more extreme ranges. While this cuts down some of the increase in the middle TC ranges it dramatically increases the overall betting efficiency. You have a 96% efficiency with hi-lo in tracking the changes in the basic strategy edge, but with playing indexes you have at least a 96% efficiency in tracking gains that simply do not drop off the same way. This results in a higher efficiency combined compared to the efficiency in predicting basic strategy edge alone!

Bottom line:
The FA is bogus; Don�s SAFE sucks; and you should follow T-Hopper�s and my advice and learn more indexes and seek higher PEs to try to profit from the bow-effect!

All true counts tend to stay the same, and the means of subsequent true counts tend to match the initial true count.

Version given in his final paper is that the true count is not changed by the exposure of more cards.

The most general version is used in my post: that any observed mean of a population will match the mean of supsequent observations of a population. See Wolfram's World of Math linked from bjmath.com See also on groups.yahoo.com/group/blackjackcardcounterscafe several of ZenGrifter's posts on knowing Abdul Jalib personaly!

pack, is about 3.2% over just 1/4 deck of depletion, for a perfect "full pack" mean pack to form another perfectly mean pack of smaller size. The probability of such being formed from an other than mean pack in 1/4 deck of depletion is far less. The chance of a meaningly significant exception causing any subset having an average of its means totalling a mean that is closer to a smaller perfect mean pack is meaninglessly small. It is totally irelevant if you consider how such a perfectly mean pack is still likely to lag as described in my main post in this thread.

The only mean that is conserved is only the mean of the largest perfect mean subpack, that is actually found in a sample, whether that mean deck is the full pack perfect mean or the perfect mean for some true count. That this will be different than the mean for a full complete pack is an absurdly small probability. It is absurd to consider anything less than the full orginal pack as the predicted mean for all subsequent subpacks.

This absurd claim that the smaller mean packs are truly representative for packs that result from a larger original pack can only be invoked to hide the type of rounding errors I have described in regard to the floating advantage.

What are you trying to prove, in plain English?

...there is a deflection in the neutral card's density,
and this density varies according to depth of penetration?

When you were done with all that verbage in all the posts you conveyed nothing except the nature of statistics is inexactness which we all knew in the first place.

These distributions talked about are normal. It makes no difference ifany outcome in a set of data conforming to the normal distribution hits directly on the mean because all the mean is is the point around which data points cluster. More points will be closer to that point (and therefore more like that point mathematically) than any other point. Looking at the mean gives one the best view possible of the import of all the data involved. Nothing more, nothing less. Looking at the standard deviation gives a feel for degrees of departures from the representative data point, the mean. The dispersal for blackjack true counts is pretty tight. If 5/6 are dealt, the true count of the last 52 cards will not have a standard deviation of more than two, meaning about 70% of possible distributions will be reasonably similar to the mean, the perfect single deck.

But let us attack this matter in a practical way. I have done numerous simulations finding invariably the expectation for a specific true count of samples of 52 cards of that true count has a higher expectation than 104 cards of that true count. The sims have shown the same higher expectation irrespective of the way the cards were stacked.

All you need to do to disprove what I have been saying is to demonstrate two deck compositions of the same true count where the expectation off the top of the 104 stack is larger than or equals the expectation of the 52 card stack. There are literally billions of compositions you can choose from to find such compositions. If you can demonstrate that for any possible true count or for any possible permutation of cards, please do so. If that composition does, in fact, sim out to be equal or larger I will certainly confirm your results and say my theories on this matter are inaccurate.

But if you cannot find a combination in the literally billions of possibilities which does not demonstrate what you are saying, please spare the board the doubletalk to which you are resorting.

The proof is very simple: the mean expectation of any deck composition is the same mean expectation of all deck compositions that can result from that original deck composition. This is just the strong general version of the True Count Theorem. In order for a segment of those future decks to have a mean that is that of a smaller complete pack it has to actually come to some smaller complete pack. The probability of that complete pack occuring is amazingly small. And the reason is the fact that the mean in any distribution, normal, cards, bad puns, is a very small proportion of that distribution.

The total of all means then comes to the mean of the complete pack: no floating advantage.

So all of your blather about what you find in a true count of +10 and -10 in complete mean packs of whatever size is meaningless. Now there is one more proof I would like to post on why there is no FA. It is especially intended for Chem nerd, Quark, Steephan, and others. But please consider why Gwynn and Seri reversed themselves on the FA at about the time DS was coming up with his first version of "SAFE play."

Toss 7s and grab 9s. This is not tracked by the hi-lo count.

Also for a pure example of the bow-effect, trade aces for tens, or tens for aces. Such a deck will prefectly have higher expectations in middle range counts and lower expectation in more extreme counts. Your version of SBA should be well suited for this. You can even derive a very detailed proof of the entire bow-effect simply from showing that the general effect of such "trades" is to change extreme and middle range edges in this way. This can be done by showing how the tens for aces trading covers the effects of all high card, which it does, and then simulate the similar effect that comes from increased pair splitting--just one effect of about 4--with low card trades and appeal to the general means arguments given above to demonstrate that it happens with all such low card trades. Easy! It would be insulting to really detail this if you really are coming to some understanding ---you have baited pretty absurdly before.

You forget that each of your mean decks is rarer still by reason of the accumulated variance from originating from a larger initial pack, also. Look above, in my post and note my statement, on how if a deck is not at some perfect mean configuration, at some 1/4 deck point, it will be far less likely, than a perfect mean composition deck, to reach any such perfect mean at the next 1/4 deck point.

Give me any concrete example, TC from -infinity to +infinity, any mixture of cards numbering 104 and any mixture of cards numbering 52 of the same true count where the expectation of the 104 cards is greater than the 52 cards having the same true count. Switch 7's and 9's to your hearts content. Switch aces and tens. Make twenty-four your low cards for the 104 card distribution fives and make none of your 52 card distribution fives.

Of the billions of possibilities demonstrate one instance. That is all I am asking.

Take virtually ANY true count for the larger pack, with normal density of aces and tens. Now take a smaller pack, with the same true count, except now skew the number of aces and tens. VIRTUALLY ALL OF THEM will exhibit, lower edges with bow effects. You can take ANY true count with the larger pack having a more normal density, and add smaller and neutral cards to example various true counts, and you will find:

an overall less edge, and

an overall bow effect.

The whole principle is that the added variation in deck compositions that is caused from origination with a larger initial pack, ABSOLUTELY ALWAYS makes the mean deck composition less likely than it would be if you started with a smaller initial pack size. Absolutely every deck drawn from the starting pack size has mean edges that add up to the starting pack, and never any smaller complete pack mean unless an ACTUAL smaller complete pack has been reached. THEN AND ONLY THEN WILL ANY SMALLER PACK HAVE MEAN EDGES THAT AVERAGE THAT OF ANY SMALLER COMPLETE PACK.