John Lewis: Comp dependent 16 v 10 indices

[John Lewis]

We know from Schlesinger's work (Blackjack Attack 2nd ed., Table 5:1 ["Illustrious 18"]) that 16 vs 10 is a singularly important hand, comprising 3.5% of all hands.

Zenfighter recently communicated self-formulated 2-card SD S17 High-Low composition dependent indices for 16 v 10.

Note that Wong's published (PBJ) SD (S17 and H17) index for all hands 16 v 10 is +1.

Zenfighter's 2-card composition dependent indices are as follows:

10, 6 v 10: +3.6
9, 7 v 10: +0.3

One would presume that 10,6 v 10 hands are almost precisely 4 times more incident than 9,7 v 10 hands, given that 10's are 4 times more incident than 9's while 6 and 7 incidences are identical.

Given Wong's net index of +1, this suggests that the the 3-or-more-card composition indices of this hand are predominantly negative, and possibly significantly negative.

It would be very interesting to examine a 3-card and 4-card composition dependent analysis of this hand.

Unable to perform these calculations myself, I hope Zenfighter or another mathematically capable responder will share my interest.

These refined indices for this important hand would be worthwhile for many serious SD players.

Thank you.

[Zenfighter]

Zenfighter's 2-card composition dependent indices are as follows:

10, 6 v 10: +3.6
9, 7 v 10: +0.3

Not exactly, they are 3.4 and -0.3, which are pretty the same as Wong?s 4 and 0, anyway.

One would presume that 10,6 v 10 hands are almost precisely 4 times more incident than 9,7 v 10 hands,

Almost yes

0.014479638009/0.00386123680242 = 3.75 by rounding

It would be very interesting to examine a 3-card and 4-card composition dependent analysis of this hand.

There are only 29 cases of 4 card 16s. Maybe Cacarulo is going to bite on this academic finesse. :-)

See the Theory page for the 3 cards, 16 vs T.

Sincerely

Z

[Cacarulo]

> We know from Schlesinger's work (Blackjack
> Attack 2nd ed., Table 5:1 ["Illustrious
> 18"]) that 16 vs 10 is a singularly
> important hand, comprising 3.5% of all
> hands.

Actually, that number is true for 6D (3.43%) but for 1D it goes up to 3.73%.

> Zenfighter recently communicated
> self-formulated 2-card SD S17 High-Low
> composition dependent indices for 16 v 10.

> Note that Wong's published (PBJ) SD (S17 and
> H17) index for all hands 16 v 10 is +1.

There are many errors in Wong's published indices so I wouldn't rely a 100% on them. The correct index for this play is "0".

> Zenfighter's 2-card composition dependent
> indices are as follows:

> 10, 6 v 10: +3.6
> 9, 7 v 10: +0.3

I believe ZF is using EORs for calculating these indices. The problem with EORs is that sometimes they give incorrect indices. By simulation I get:

T6 vs T = +3
96 vs T = +0

Doing an algebraic analysis I also get +3.6 and -0.3 but I know +3.6 is not right because using a more exhaustive simulation the index is still +3.0 (with one decimal point of precision).

> Given Wong's net index of +1, this suggests
> that the the 3-or-more-card composition
> indices of this hand are predominantly
> negative, and possibly significantly
> negative.

3-or-more-card hands appear 1.9% of the time.

> It would be very interesting to examine a
> 3-card and 4-card composition dependent
> analysis of this hand.

I don't think this would be worth learning. In these cases it is probably better to use the info provided by CA than to use indices. This is: STAND with 3 or more cards. There are some finer points to this that you can learn in order to increase efficiency.

> These refined indices for this important
> hand would be worthwhile for many serious SD
> players.

The only indices that matter here are T6 and 97.

Hope this helps.

Sincerely,
Cacarulo

[Cacarulo]

> Zenfighter's 2-card composition dependent
> indices are as follows:

> 10, 6 v 10: +3.6
> 9, 7 v 10: +0.3 Not exactly, they are 3.4
> and -0.3, which are pretty the same as
> Wong?s 4 and 0, anyway.

As I said in my previous post the indices should be +3 and 0. Besides, EOR-based indices do not take into account penetration and/or cut-card effect.

> It would be very interesting to examine a
> 3-card and 4-card composition dependent
> analysis of this hand. There are only 29
> cases of 4 card 16s. Maybe Cacarulo is going
> to bite on this academic finesse . :-)

Be careful on how you count the cases. You should count "all" the possible hand combinations. As an example take the following 3-card composition:

556 vs T

Assuming this as ONE group comprised of two fives and one six would imply that any combination of these three cards is equally likely to occur which is obviously wrong. 655 and 565 should never occur because we would be doubling 11 v T (American rules)

Having said this I get:

2-card combinations:    4 cases (T6,6T,97 and 79) 

3-card combinations:   60 cases 

4-card combinations:  283 cases 

5-card combinations:  708 cases 

6-card combinations: 1012 cases 

7-card combinations:  834 cases 

8-card combinations:  307 cases 

9-card combinations:   15 cases

Sincerely,
Cacarulo

[John Lewis]

The 3-card hand analysis of 16 v 10 is indeed interesting, and, in my opinion, useful to the player. I have posted a proposed summary of your data on Theory page.

Thank you, JL

[John Lewis]

Cacarulo --

Thank you for your input on this question. I am a great admirer of your work.

"Be careful on how you count the cases. You should count "all" the possible hand combinations. As an example take the following 3-card composition:

556 vs T

Assuming this as ONE group comprised of two fives and one six would imply that any combination of these three cards is equally likely to occur which is obviously wrong. 655 and 565 should never occur because we would be doubling 11 v T (American rules)"

-- great point

"Actually, that number (16 as 3.5% of all hands) is true for 6D (3.43%) but for 1D it goes up to 3.73%."

-- another great point

"There are many errors in Wong's published indices so I wouldn't rely a 100% on them. The correct index for this play is "0"."

-- thanks for the information

"[" It would be very interesting to examine a 3-card and 4-card composition dependent analysis of this hand."]

I don't think this would be worth learning. In these cases it is probably better to use the info provided by CA than to use indices. This is: STAND with 3 or more cards. There are some finer points to this that you can learn in order to increase efficiency."

-- It would indeed be excessively laborious to learn each 3 card 16 v 10 index individually. In my post on Theory page, however, I have proposed a simplified approach for accomodation and use of the 3 card indices.

"3-or-more-card (16 v 10) hands appear 1.9% of the time."

-- nice number, thanks

-- what % of the time do 4 card hands appear?
-- what % of the time do 4-or-more card hands appear?

Thanks, JL

[Cacarulo]

> Thank you for your input on this question. I
> am a great admirer of your work.

Thank you! You're welcome.

> "[" It would be very interesting
> to examine a 3-card and 4-card composition
> dependent analysis of this hand."]

> I don't think this would be worth learning.
> In these cases it is probably better to use
> the info provided by CA than to use indices.
> This is: STAND with 3 or more cards. There
> are some finer points to this that you can
> learn in order to increase efficiency."

> -- It would indeed be excessively laborious
> to learn each 3 card 16 v 10 index
> individually. In my post on Theory page,
> however, I have proposed a simplified
> approach for accomodation and use of the 3
> card indices.

Yes, I saw your post and it's really very interesting. I haven't thought of using indices in that way. If we can adjust those frequencies they would be worth learning!

> "3-or-more-card (16 v 10) hands appear
> 1.9% of the time."

> -- nice number, thanks

> -- what % of the time do 4 card hands
> appear?
> -- what % of the time do 4-or-more card
> hands appear?

2+ = 3.73% 

3+ = 1.90% 

4+ = 0.47% 

5+ = 0.06% 

6+ = 0.004% 

7+ = 0.0001% 

8+ = 0.00% 

9+ = 0.00%

So, exactly 4 cards can be calculated as 0.47% - 0.06% - 0.004% - 0.0001% = 0.41% (approx.)

Hope this helps.

Sincerely,
Cacarulo

[Cacarulo]

> It would be very interesting to examine a
> 3-card and 4-card composition dependent
> analysis of this hand. There are only 29
> cases of 4 card 16s. Maybe Cacarulo is going
> to bite on this academic finesse . :-)

Indeed there are 29 cases of 4 card 16s but, as I said, we have to distinguish the order of the hands. Actually, there are:

2 cards =  2 cases 

3 cards = 15 cases 

4 cards = 29 cases 

5 cards = 35 cases 

6 cards = 31 cases 

7 cards = 20 cases 

8 cards =  9 cases 

9 cards =  2 cases

Sincerely,
Cacarulo

[Cacarulo]

I made a mistake in the following table:

> 2+ = 3.73%
> 3+ = 1.90%
> 4+ = 0.47%
> 5+ = 0.06%
> 6+ = 0.004%
> 7+ = 0.0001%
> 8+ = 0.00%
> 9+ = 0.00%
> So, exactly 4 cards can be calculated as
> 0.47% - 0.06% - 0.004% - 0.0001% = 0.41%
> (approx.)

The numbers are a little lower:

2+ = 3.44%  

3+ = 1.76% 

4+ = 0.44%

So, exactly 4 cards can be calculated as
0.44% - 0.06% - 0.004% - 0.0001% = 0.38% (approx.)

Sorry.

Sincerely,
Cacarulo

[Cacarulo]

> Actually, that number is true for 6D (3.43%)
> but for 1D it goes up to 3.73%.

As I said somewhere it's not that up. It goes up to only 3.44%.

Sincerely,
Cacarulo

[John Lewis]

Cacarulo --

Thank you for your continued feedback and for the additional data you have supplied.

"" -- It would indeed be excessively laborious to learn each 3 card 16 v 10 index individually. In my post on Theory page, however, I have proposed a simplified approach for accomodation and use of the 3 card indices." -- JL

Yes, I saw your post and it's really very interesting. I haven't thought of using indices in that way. If we can adjust those frequencies they would be worth learning! " -- Cacarulo

---- Your post on Theory shows you're ready to do the necessary frequency adjusted evaluation. I'm not going to try it on my calculator if you are set up to do it on your computer!

I just posted on Theory a simplification which presumably will be the preferred 3 rules for management of the 16 v 10 using 2-card or 3-card comp dependent indices. Unless, of course, your frequency adjusted data changes everything.

""-- what % of the time do 4 card hands
appear?
-- what % of the time do 4-or-more card
hands appear?" -- JL

2+ = 3.73%
3+ = 1.90%
4+ = 0.47%
5+ = 0.06%
6+ = 0.004%
7+ = 0.0001%
8+ = 0.00%
9+ = 0.00%

So, exactly 4 cards can be calculated as 0.47% - 0.06% - 0.004% - 0.0001% = 0.41% (approx.) " -- Cacarulo

---- Good data, Cacarulo, thank you. Four card 16 v 10 totals represent only 1/7th of the entire hand, but yet they still comprise 1/200 of all hands. Still a signficant consideration.

I continue to believe it would be helpful to examine 4-card 16's before the rules for comp dependent indices for this most important hand are decided and no longer examined.

"" It would be very interesting to examine a 3-card and 4-card composition dependent analysis of this hand." -- JL

I don't think this would be worth learning. In these cases it is probably better to use the info provided by CA than to use indices. This is: STAND with 3 or more cards. There are some finer points to this that you can learn in order to increase efficiency." -- Cacarulo

---- It seems we may have agreed that examination of the 3-card indices for this hand may be worthwhile. Let's check out the 4-card data, too -- please.

Thanks, JL

[John Lewis]

[Cacarulo]

> I continue to believe it would be helpful to
> examine 4-card 16's before the rules for
> comp dependent indices for this most
> important hand are decided and no longer
> examined.

> ---- It seems we may have agreed that
> examination of the 3-card indices for this
> hand may be worthwhile. Let's check out the
> 4-card data, too -- please.

It can be done but you can't apply to 4-card 16s the same rule (hit if there is a 6, otherwise stand). That rule is only valid with 3-card 16s. So first you need to find a new rule for this case. Overall, this is not worth it since there are few occasions in which hitting is better than standing. The best rule is to always stand.

Sincerely,
Cacarulo