John Lewis: 3 card CD index for 12 v 4?

[John Lewis]

12 v 4 is a very significant hand, a member of the Illustrious 18. 2 card CD indices for the hand are well known. A more detailed examination of the hand (a mean 3 card index, for example) has not been performed, to my knowledge.

The 2 card CD indices (SD) of the hand are as follows (Professional Blackjack; Zenfighter): 10,2:+2; 9,3:0; 8,4:0; 7,5:0. The mean index of the hand is 0.

Note that 10,2 is a much more common hand than the other 2 card indices.

Is it possible that the mean 3 card index is a negative one, since the mean index is 0?

-2 or thereabout, perhaps?

If so, this would be a valuable CD index.

I hope someone on the site shares my interest.

JL

[Zenfighter]

SD, S17

 

 

Hand	        Index		 m(1)	 	Action 

 

2,3,7 vs. 4	0.7		-3.86		stand 

 

2,4,6 vs. 4	1.9		-3.71		stand 

 

2,5,5 vs. 4	1.9		-3.66		stand 

 

3,4,5 vs. 4 	1.2		-4.68		stand 

 

 

Moral? Do not hit a three-card holding vs. a 4 in SD. The full-deck favorability for hitting is averse. Use just common sense.

Hope this helps.

Zenfighter

[Don Schlesinger]

> Is it possible that the mean 3 card index is
> a negative one, since the mean index is 0?

> -2 or thereabout, perhaps?

I know what you were thinking, but the analysis is incorrect. As ZF has shown, the correct index, depending on the holding, is usually 0 or +1 (floored). That's considerably lower than the TC you would have simply by receiving the three cards that add to 12 and dealer's 4, but it isn't the negative value (-2) that you mention, because we start with +4, just by receiving the hand.

Your mistake is thinking that the count is zero, to begin a BS play; it isn't. In this case, it's often +4.

Don

[John Lewis]

Zen

Thank you for the interesting data.

I was initially surprised by the results. Again, I reasoned that the mean 3 card index would be negative to bring the mean overall hand index to 0 in light of a 10,2 CD index of +2.

Perhaps the simple flooring of the other 2 card combinations (by raising them) as well as the flooring of the mean index (by lowering it) accounts for the discrepancy.

I would like to make a few (admittedly, very esoteric) comments on this matter, but please do not surmise that I am suggesting that your sample is an inadequate one to address this issue. I am not suggesting that.

Comments:

1) Not all 3 card hands are represented here, of course. 2,2,8; 3,3,6; 4,4,4 (certainly too rare in SD against a dealer 4, to be of consequence); A,2,9; A,3,8; A,4,7;and A,5,6 are others obvious to me.

The ace containing hands, in particular, might have a lower index simply because of the hilo idiosyncracy/deficiency that counts ace as -1 for playing strategy in this instance where aces dealt and thus out of play, though lowering the count, really do not make hitting more desirable (whereas the removal of the more appropriately valued 10 cards does make hitting more desirable).

Let me give an example to illustrate this point. At a starting count of -2 (head to head), a player is dealt an 8,4 v dealer 4. The count at the time of hit/stand decision is thus 0, and the player would correctly stand. But, say the player is dealt not 8,4, but A,3,8 v the dealer 4. The count is now -1, not 0. The player, using the mean index, would hit. But this would be despite the fact that the risk of busting by hitting an A,3,8 is not appreciably less than that of hitting an 8,4. And we know well that stand on 8,4 v 4 at 0 count is the correct play. Thus the CD index of A,3,8 is presumably -1 vs 0.

This misrepresentation of the effect of the ace on playing strategy in hilo is a phenomenon common to all stiff hands (as well as, notoriously, double on 11). This phenomenon would likely be better addressed, if desired, by appropriate use of an ace side count than a CD index.

2) The mean index would be affected by 4-or-more card hands, also. But one would not expect these hands to occur frequently enough to have a significant effect.

3) It's curious to note that the playing strategy of the majority of these 3 card combinations are deal order dependant. Many yield initial 2 card totals of 9, 10, or 11 when dealt in random order. In these instances, in all but quite negative counts, the first two cards would prompt a double down against a 4 vs. a hit. Similarly, the ace containing hands often yield soft 17-20 on the first two cards dealt, which would almost invariably not call for a hit against dealer 4 (the sole exception would be A,6 in very negative counts). An initial pairing of 2's and 3's vs. 4 in neutral or positive counts would reduce the frequency of participation of hands containing these cards in a 12 total also, since the pair would be split rather than hit. All of these non-hit first two card totals would make the contribution of 3 card combinations to the 12 v 4 hand less frequent and thus of much less significance than one would offhandedly expect.

Conclusion: 3 card player 12's v dealer 4 almost certainly do not present anything approximating a single coherent index, and neither do they merit consideration as a more fragmented group of CD indices.

Thanks, again.

John

[John Lewis]

"I know what you were thinking, but the analysis is incorrect. As ZF has shown, the correct index, depending on the holding, is usually 0 or +1 (floored). That's considerably lower than the TC you would have simply by receiving the three cards that add to 12 and dealer's 4, but it isn't the negative value (-2) that you mention, because we start with +4, just by receiving the hand." -- DS

Don

Thanks for your comment.

Your point is, I believe, that 3 card player 12's v dealer 4 would, in the great majority of instances, present a playing strategy decision in a positive count, given the effect on TC that the hand itself creates in the SD game. Only if the beginning (pre-deal) count were significantly negative would this hand likely be a candidate for a hit decision. Thus the frequency of occurrence of 3 card 12's that would actually utilize this index would be accordingly small.

Yours is yet one more argument that 3 card index considerations for this hand are not worthwhile.

JL

[Zenfighter]

1) Not all 3-card hands are represented here, of course. 2,2,8; 3,3,6; 4,4,4 (certainly too rare in SD against a dealer 4, to be of consequence); A,2,9; A,3,8; A,4,7;and A,5,6 are others obvious to me.

Note that for SD, S17 and DAS as per table B2 from BJA3 all this hands do not exist, because the use of basic strategy preclude their existence. I assumed this and thus the limited sample.

Ditto for their reversals also: 6,5,A; 7,4,A; 8,3,A; 9,2,A, where a double down is called for. The same with 4, 4 vs. 4. Here you have a split.

Sincerely

Zenfighter

[Don Schlesinger]

> Yours is yet one more argument that 3 card
> index considerations for this hand are not
> worthwhile.

I had no doubt whatsoever, from the inception of the discussion, that three-card C-D indices were not "worth" learning. You can take that as a given. But, this is, after all, the Theory & Math page, so pursuing the math is/was an interesting, if not practical, exercise, no?

Don

[John Lewis]

[John Lewis]

> I had no doubt whatsoever, from the
> inception of the discussion, that three-card
> C-D indices were not "worth"
> learning. You can take that as a given.

Sometimes an analysis will surprise you, but you were certainly right in this instance (big surprise there).

> But, this is, after all, the Theory & Math
> page, so pursuing the math is/was an
> interesting, if not practical, exercise, no?

I agree with that.

At least our discussion explained that 3 card totals of the smaller stiffs are infrequently seen, and why that is so.

On the site last year, upon examination of the 16 v 10 index, the disparity between the 2 card CD indices and the mean index did expose a surprisingly uniform 3 card index that was otherwise unsuspected.

Examination of the 16 v 9 indices yielded an similar finding.

The present examination showed us that this technique of discovery has limited rather than universal utility.

Thanks again for the immediate feedback. That's the beauty of this site (accessiblity and responsiveness of the expert hosts).

JL

[Zenfighter]

Composition dependent indices for double decks?

Peter Griffin: The method of computing the gain from learning two-card indices as opposed to learning single non-composition dependent index for any decision, is to calculate the correlation of the count system for each two-card to obtain a weighted average correlation for the decision, comparing this to the correlation of the non-composition dependent decision.

An example shall suffice (2 dks, s17)

 

Hand		Correlation	Generic correlation 

 

T,2 vs. 4	.771668		.771539 

9,3 vs. 4	.77271 

8,4 vs. 4	.772146 

7,5 vs. 4	.773222 

 

Here the weighted correlation:

wc =(Sum c(i) * p(i))/ 4 {for i = 1 to 4}, yields:

wc = .772106

Here the p(i) for 2 dks as per Cacarulo?s EV tables.

That is: you have increased the correlation of your point count system to handle the hand by an:

Delta = 0.0735%

Now lets find the Hilo gain using the generic index at the 52 card-level.

Gain (g) = 2.44101%, thus we can predict and increase by using comp-dependent indices:

Gain (cd) = 2.44280%

Watch e.g. a $50 bet and the hand in question:

Gain(g) = 50 *2.44101% = $1.220505

Gain (cd) = 50 * 2.44280% = $1.221400

That is, you have won here for your increased memory efforts a total of:

Total won = $0.000895

So, we have learned that such strategies are not practical at all, because your correlation will not increase sufficiently so as to expect any noticeable increase in your earnings.

Don would probably add: ?I will bite on this: a couple of hot dogs per year.? :-)

Let?s be realistic, then.

Hopefully this short example will help you understand my reluctance to work through all of them.

Sincerely

Zenfighter

[John Lewis]

Zen

Thank you for the communication. Our discussion of this started with a private communication. I think it appropriate to post that communication at this time to put your comments in context:

2/10/05: Zenfighter

I have compared the SD indices I summarized from the data you provided me last year (thank you once again) with Wong's published 4D CD's. I have crudely interpolated some results for DD. I believe several of these DD indices would be worthwhile to actually employ in that game.

The 16 v 9 and 16 v 10 indices are again categorized the way I summarized and categorized the SD CD indices that you (and Cacarulo) provided me. All of those results were posted on AP.com in 12/03 and 1/04, as you may recall.

The following are what I have speculated to be the significant DD CD indices:

10,6 and 3-card 16's containing 6 (except 6,5,5): +2
9,7: 0
3-card 16's not containing 6 (except 6,5,5): -1

This hand (16 v 10) is so valuable that I believe any fastidious DD counter might benefit from these indices, if they are indeed valid.

Continuing:

2-card 16 v 9: +6
3 or more card 16 v 9: +3 or +4

10,3 and 9,4 v 2: +1
8,5 and 7,6 v 2: -1

10,2 v 2: +4 or +5
7,5 v 2: +2

10,2 v 3: +3
7,5 v 3: +1

9,2 appears to be approx -3, only minimally different from the mean index of -4; 6,5 however could be either -2 or -3.

John

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

Now back to your present post.

I am not aware of the availability of DD CD indices. The fact that certain CD indices remain separate from other CD indices of their parent hand at 4D (Wong) suggests the existence of valid CD indices at DD, however.

Using the same crude interpolative technique by which I arrived at the above suggested indices, I suggested that the 10,2 v 4 index at DD might be +1, though the mean index is 0 (bj21.com). You were kind enough to run the numbers on this, and indeed you demonstrated that this index is indeed +1, as predicted.

Your present post demonstrates a real, yet quite small, benefit by employing this single index.

I remain interested in the examination of the other putative DD CD indices I suggested to you.

Some of that interest is, admittedly, "academic." In other words, I don't expect a tremendous playing benefit to arise from these specialized indices. But the discovery, validation, and mathematical evalution of gain provided by these indices remains a matter of curiousity.

But, also, even if these indices are of minimal benefit, if they are validated as correct I intend to use them. I use the SD DD indices, several of which we arrived at on this site, routinely. The slight modification and use of these same indices at DD will be quite simple and effortless.

Thank you, once again, for your attention, and thank you for introducing this matter we have privately discussed onto the AP discussion board.

JL

[Zenfighter]

Your present post demonstrates a real, yet quite small, benefit by employing this single index.

Make no mistake here. Actually both figures are in a sense not real. In other words I?ve used a laboratory example assuming you have already a 12 vs. 4 in your hand, the TC = 0 and you have a $50 on the betting circle.

The problem is that your real gain will be the difference between both, adjusted for their a priori probability of occurrence of both the hand and the TC = 0 while playing a double-deck game at this selected card level (52 cards remaining). Meager figures will be the final calculations. Table 5.1 from Don?s IL18 is enough prove of the complexity of the calculations. Trying to duplicate something like this is really demoralizing, for the amount of task involved. Imagine cycling through all the indices. God forbid! The fact will remain untouched anyway, and that is, that your playing correlation increased value do not point in the direction of accomplishing such a giant effort, like is, to calculate with precision how much you?ll gain exactly with every index. .

A lot easier seems to me to extract a full comp-dependent set of indices for two-decks, so that you?ll be able to use it at your own risk. You?re forewarned now.

What are the exact rules you?re interested in? Mainly h17 or s17?

Best regards

Zenfighter

[John Lewis]

"That is, you have won here for your increased memory efforts a total of: Total won = $0.000895" ($50 bet)-- Zenfighter, 2/18

Zen:

Yeh, but what about with a really fast dealer?

That comment was actually self-directed sarcasm. I'm actually not quite that out of touch -- but, perhaps, almost.

Finally, the significance of your figures filtered through my thick brain. We're talking a thousandth of a cent on a $50 bet ON THIS PARTICULAR HAND! A cent on $50,000 bet ON THIS (12 v 4) HAND! Is that right?

I can see why you discuss the possibility that this gain may not even be "real." This is like a quantum mechanical treatment of blackjack. At my request, you have approached the quantum realm, where values are never certain. That one-thousandth of a cent flickers in and out of existence like elementary particles spontaneously appear and disappear in a vacuum.

In other words, sorry for the dumb questions.

If I had correctly processed the gain information you posted on 2/18 (the post before your last), I would have apologized at that time for asking all these questions about matters of almost total inconsequence. My follow up post would not have been one that suggested an even more detailed exploration of the matter.

So, please, "excuse me."

You and Don both cautioned me last year that the significance of these indices was very minimal even at SD. So I should have accepted that about DD. Your calculation of actual gain from 12 v 4 CD indices certainly settles the matter.

This makes one wonder: given the profound insignificance of these CD indices at 2D, why did Stanford Wong publish CD indices for 4D in Professional Blackjack? Both S17 and H17, at that?

"Table 5.1 from Don?s IL18 is enough prove of the complexity of the calculations. Trying to duplicate something like this is really demoralizing, for the amount of task involved. Imagine cycling through all the indices. God forbid! " -- Zenfighter

I apologize for my naivete in these matters. When I speciously ask mathematical outcomes of arcane issues, I am somewhat like a child in a toy store that wants everything in the store but has no idea of practical realities. And this realization actually preceded the realization of just how insignificant my questions were!

You are a prince of a guy to have been as courteous and accommodating as you were through this entire thread. I'll try to keep the dumb questions to a minimum from now on.

John