Frequency Distribution of Number of Cards Dealt Heads Up

[Peter JF]

I have tried to get the answer to this from an existing thread answering an approximate question on this, which is simply the average number of cards at about 5.4, or slightly less heads up due to player BJs reducing the number of cards a dealer will deal themselves.

The thread I refer to is https://www.blackjackinfo.com/community/threads/how-many-cards-per-hand-average.15226/#post-496208

However, what I am looking for is something like the average percentages as follows, and up to 15 dealt cards:

4 cards (pat hand with dealer 17 or above with 2 cards): 15%
5 cards (any combination of 2 and 3 between dealer and player): 20%
6 cards (any combination of cards between dealer and player): 15%
7 cards (any combination of cards between dealer and player): 12%
8 cards (any combination of cards between dealer and player): 10%
9 cards (any combination of cards between dealer and player): 7%
10 cards (any combination of cards between dealer and player): 6%
11 cards (any combination of cards between dealer and player): 5%
12 cards (any combination of cards between dealer and player): 4%
13 cards (any combination of cards between dealer and player): 3%
14 cards (any combination of cards between dealer and player): 2%
15 cards (any combination of cards between dealer and player): 1%

I think this information will be available to someone with a calculator and that can set up a heads up game.
It will be interesting to see how this varies with different number of decks, it much at all.

For sake of clarity on rules we can assume DAS, S17 and any other typical Las Vegas rules. Maybe some variation on rules could be assessed to see if this has any significant impact or not.

Hopefully someone has the software to answer this question.

 

[DSchles]

I asked once before, and I'll try again: is there a practical reason why you want this information? People are reluctant to do the work if they see no reason for wanting the information. While no one can require you to give it, it might help you to get an answer.

Don

 

[xengrifter]

I asked once before, and I'll try again: is there a practical reason why you want this information? People are reluctant to do the work if they see no reason for wanting the information. While no one can require you to give it, it might help you to get an answer.

Concur.
Peter, please describe WHERE you are going with your request, your purpose or end goal...
...or are you just curious?

 

[Peter JF]

I asked once before, and I'll try again: is there a practical reason why you want this information? People are reluctant to do the work if they see no reason for wanting the information. While no one can require you to give it, it might help you to get an answer.

Don

Sorry I thought I had answered this on the last thread... it was for reasons of looking at a count estimating system. Maybe something that is easier for people to use than normal counting, but to do the checks on it I need to know this frequency distribution to see if it a non-starter or has some mileage in looking at it further.

 

[Peter JF]

Concur.
Peter, please describe WHERE you are going with your request, your purpose or end goal...
...or are you just curious?

Hi Zengrifter, see reply to Don. Or if you know of any software that can do this then that would help and I can get it and run the simulations. Thanks, Peter

 

[Peter JF]

Concur.
Peter, please describe WHERE you are going with your request, your purpose or end goal...
...or are you just curious?

Hi Zengrifter, see reply to Don. Or if you know of any software that can do this then that would help and I can get it and run the simulations. Thanks, Peter

 

[DSchles]

Hi Zengrifter, see reply to Don. Or if you know of any software that can do this then that would help and I can get it and run the simulations. Thanks, Peter

Not clear why the average number of cards used per round, globally, isn't sufficient for your needs. What's the difference how that average breaks down?

Don

 

[xengrifter]

Not clear why the average number of cards used per round, globally, isn't sufficient for your needs. What's the difference how that average breaks down?

It sounds to me like Peter is on the verge of re-discovering a "SpeedCount" type system.

 

[DSchles]

It sounds to me like Peter is on the verge of re-discovering a "SpeedCount" type system.

Yes, sadly, that's exactly right. But, if so, knowing the breakdown by hand types is completely unnecessary.

Don

 

[Peter JF]

Thanks for the heads up on the re-invention aspect as nobody wants to be involved in that due to being a waste of time. I did some research into that term at least and it did come up with something based on the average. However, its not that but it is an estimator, I would hope much more accurate and less prone to drifting away from the average. I don't think I am going to get my answer here unfortunately which is a shame as really expected this to be something an existing calculator could answer. If anyone knows of something that can help here please let me know.

 

[Dummy]

It sounds like he is going to count the number of cards in the dealer's hand so it will infer the count for his next bet. lol

 

[Peter JF]

It sounds like he is going to count the number of cards in the dealer's hand so it will infer the count for his next bet. lol

No, it is no that Dummy, but very funny. I see people like fun on this site, and in their books in this part of the forum. I like that... often with a dry sense of humour. Anyway, but way of general update, I was working on this last night after suffering bad jet-lag on return from Vegas (I am from the UK as you can tell with the spelling), I used Norms table for card lengths dealt to the player and then adopted this as an approximate of distribution of card lengths dealt to the dealer (at least the average is the same and I know it comes down closer in heads up play for obvious reasons).
Well the outcome was that the principle seems good and the figures are in the right region for a good approximation to the real count. I used it last week in Vegas and it worked well (both in terms of anticipation of a rich part of the deck arriving, as well as getting me to walk from the table when that came but that cards didn't... just in case there was another reason for a 'favourable' count arising towards the end of the shoe!).
The good thing was that I did not really have to 'learn' it by practicing many hours, I just had to concentrate on what I was doing (not always easy for me). So it worked from the off after developing the count on a basic spreadsheet.
I think the next step is to find someone to work with to develop this further if there is any interest from someone wanting to do this with me as I don't really have the time or ability to get all the stats like I would like to prove it to others, as well as prove it is not a re-invention.

 

[gronbog]

You can not have played even close to enough rounds over the course of one week to draw any statistically valid conclusions. You need to simulate the system. When you're ready with your final system, contact me. I can do this for you.

 

[DSchles]

No, it is no that Dummy, but very funny. I see people like fun on this site, and in their books in this part of the forum. I like that... often with a dry sense of humour. Anyway, but way of general update, I was working on this last night after suffering bad jet-lag on return from Vegas (I am from the UK as you can tell with the spelling), I used Norms table for card lengths dealt to the player and then adopted this as an approximate of distribution of card lengths dealt to the dealer (at least the average is the same and I know it comes down closer in heads up play for obvious reasons).
Well the outcome was that the principle seems good and the figures are in the right region for a good approximation to the real count. I used it last week in Vegas and it worked well (both in terms of anticipation of a rich part of the deck arriving, as well as getting me to walk from the table when that came but that cards didn't... just in case there was another reason for a 'favourable' count arising towards the end of the shoe!).
The good thing was that I did not really have to 'learn' it by practicing many hours, I just had to concentrate on what I was doing (not always easy for me). So it worked from the off after developing the count on a basic spreadsheet.
I think the next step is to find someone to work with to develop this further if there is any interest from someone wanting to do this with me as I don't really have the time or ability to get all the stats like I would like to prove it to others, as well as prove it is not a re-invention.

Google "Speed Count."

Report back.

Don

 

[Peter JF]

Google "Speed Count."

Report back.

Don

OK reporting back after a Google search of 'speed count' Don....

Well if it is all about the average of 2.7 cards per hand (and I guess same approximately for dealer albeit 2.9), and 1.01 or so for each high or low counted card (but you only count low cards), then not that is way off the mark of the principle of the system I am looking at. In fact, my logical concern with the 'speed count', if it is the same as the one called Easy OPP too, is that the count would be significantly affected by the standard deviation in the median length of hands in cards dealt per round. By comparison, the approach I am looking at would significantly reduce that variation to make a more reliable estimated count. It is also based on a balanced count system, which again is a difference, albeit not a claimed advantage (except for use in optimal play basic strategy variation).

I also agree with Gronbog's comments above that my experience is not statistically valid when I took it to the casinos last week in Vegas. That was really more to test out whether it was as easy to implement as I expected, which it was. That is also important to prove of course.

I can see my new approach (which I can't find documented on the higher ranking pages of Google) could be subjected to a lot of statistical analysis like the Speed Count, and therefore could be compared for performance. I expect it to be better, especially if the variation from the true count figure is used as a metric. I mean the true count figure for a well-known and documented system to which it best approximates. This also means that the playing strategy by variation of basic strategy play is already documented and proven for it.

In addition, given that nobody has answered my question about the frequency distribution of the number of cards dealt (although Norman Wattenberger has this for the player-only in his book) I suspect that should support the inventive nature of this approach I am developing. I have logic behind it and some basic statistics, but nothing rigorous enough to fully validate it.

I also see that the Speed Count is proprietary which is good for the inventors and of course they deserve to be rewarded if such a system does deliver a significant profit opportunity to the investor in that system. However, it seems to be discussed elsewhere in terms of how it works and it is clear to see how it does work , with the Easy OPP system being a non-proprietary system that is supposed to work the same way. Correct me if I am wrong please.

What I am more interested in right now is whether there is anyone wanting to work on developing and proving this system for commercialisation, if the initial tests prove supportive of course. I think these will. Please contact me if this is the case with anyone reading this.

I can understand the negativity towards such a new approach, given the life-time of card counting so far and therefore limited room for any new 'discoveries'. However, the only downside that I see is that it is limited to a small game of, say, one or two players to be most effective, although with some additional skill it could be extended. However, we know the best play is heads-up with the dealer, and that is where its strengths are, which I think is no bad thing. Not surprisingly, that is my mode of play, playing the tables that are least busy and preferably heads up.

I hope that is sufficient to class as a 'report back' Don. Thanks for your interest in my post and the subject of discussion here.

 

[xengrifter]

Easy OPP system being a non-proprietary system that is supposed to work the same way.

Yes

 

[Peter JF]

The results have been done by Gronbog now. It was quite a journey because let's say the principle was right but there was something in the detail that was not working as well as expected in this new approach to card counting. However, due to Gronbog's patience with me (an understatement!), I managed to find out what the detail was that was stopping the principle from working and now we have the simulation of the system both with and without indices. We have compared this to the other most popular/well-known card counting systems to put its performance in proper perspective. The SCORE results table is below for the full card counting comparison performance details.

In summary, I would conclude from these tables that whilst it is not in the same league as REKO and HILO, it is the 'Best of the Rest'. As this was never intended to be a card counting system that would challenge the best (most people find such card counting not as easy as those who say it is easy!), I consider this a success level I am very happy with. In particular, it outperforms the Speed Count which also requires quite a high degree of concentration at the tables. In fairness, it is in let's say the same league as the Speed Count but in most cases comes out at the top under most scenarios. In fact, if used with just Basic Strategy (no play variations), then it generally performs as well as Speed Count with its Optimal Basic Strategy (another table to learn different to BS). Speed Count being a very popular 'easy' system with some well-known acclaimed degree of success.

I am not going to go into the details of how FD-27 works here. But suffice to say, as I did when I first brought this to the forum, that it is intended for Heads Up play. To that end, Gronbog has also analysed its performance in that situation and it comes out with very similar results. I make no apology for it being used in that situation, as that is when I like to play as you get more hands played per minute and therefore more profit potential... even if that means playing the tables at much less social times of the day to get the relatively empty tables in order to do this (it can tolerate another player at the table but then would lose effectiveness as more players join).

In such a situation of heads up play with the dealer, the use of the system means that counting can be economised as the cards deal effectively do the averaging of the count themselves, and then there is just one alteration to the count required at the end of each round based on the situation observed during play. There is no up and down counting like in many traditional counts, just an up or down adjustment to the running count at the end of the round.

In the table below there is also the very simple count of the Ace/Ten count, which really comes nowhere near in performance. Even Thorp's original 10-Count is blown away by the FD-27, the very count system acclaimed for turning the tables on Vegas by the pioneer mathematical card counter Edward O. Thorp.

Thanks to the support from Gronbog we got there.... I know it took a long time... but at least what was started in Easter is finished before Christmas... we are both part-time working on this of course, with full-time professions to focus on, hence we both worked in spare time on this project as time allowed. The main thing I think is that we got a positive result to prove the principle of the system and assess its performance level, which was our original objective.

Maybe this is not of interest to the skilled card counter (apart from academic interest of course), but for the person wishing to get rapid results for minimal effort as an introduction to winning at the blackjack tables, I think it has much to offer.

Results table by Gronbog follows....

SCORE 6.5/8 S17 DAS

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

Scenario System & SCORE

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

Play-All 1-8

Ace/5 No Indices 0.04

Thorp 10 No Indices 1.39

Speed No Indices 2.46

FD-27 No Indices 3.04

Speed Optimal Basic Strategy 3.08

FD-27 Full Gronbog Indices 5.05

REKO No Indices 5.53

HiLo No Indices 6.28

REKO QFIT Indices 10.34

HiLo Sweet 16 Indices 11.44



Play-All 1-10

Ace/5 No Indices 0.23

Thorp 10 No Indices 2.25

Speed No Indices 3.60

FD-27 No Indices 4.41

Speed Optimal Basic Strategy 4.43

FD-27 Full Gronbog Indices 6.73

REKO No Indices 7.67

HiLo No Indices 8.35

REKO QFIT Indices 13.49

HiLo Sweet 16 Indices 14.46



Play-All 1-12

Ace/5 No Indices 0.45

Thorp 10 No Indices 3.00

Speed No Indices 4.53

FD-27 No Indices 5.52

Speed Optimal Basic Strategy 5.54

FD-27 Full Gronbog Indices 8.03

REKO No Indices 9.38

HiLo No Indices 10.04

REKO QFIT Indices 15.99

HiLo Sweet 16 Indices 16.81



Back-Count 1-1

Ace/5 No Indices 2.94

Thorp 10 No Indices 8.22

Speed No Indices 10.31

Speed Optimal Basic Strategy 12.30

FD-27 No Indices 12.45

FD-27 Full Gronbog Indices 15.19

HiLo No Indices 18.96

REKO No Indices 19.26

HiLo Sweet 16 Indices 27.17

REKO QFIT Indices 28.39



Back-Count 1-2

Ace/5 No Indices 3.39

Thorp 10 No Indices 9.71

Speed No Indices 11.98

Speed Optimal Basic Strategy 14.28

FD-27 No Indices 14.49

FD-27 Full Gronbog Indices 18.12

REKO No Indices 22.45

HiLo No Indices 22.75

HiLo Sweet 16 Indices 33.02

REKO QFIT Indices 33.61



Back-Count 1-4

Ace/5 No Indices 3.61

Thorp 10 No Indices 10.35

Speed No Indices 12.57

Speed Optimal Basic Strategy 15.02

FD-27 No Indices 15.30

FD-27 Full Gronbog Indices 19.27

REKO No Indices 23.70

HiLo No Indices 24.01

HiLo Sweet 16 Indices 35.75

REKO QFIT Indices 35.98



Back-Count 1-8

Ace/5 No Indices 3.63

Thorp 10 No Indices 10.56

Speed No Indices 12.68

Speed Optimal Basic Strategy 15.21

FD-27 No Indices 15.87

FD-27 Full Gronbog Indices 20.24

REKO No Indices 24.08

HiLo No Indices 24.67

HiLo Sweet 16 Indices 36.61

REKO QFIT Indices 36.80



Back-Count 1-12

Ace/5 No Indices 3.64

Thorp 10 No Indices 10.59

Speed No Indices 12.71

Speed Optimal Basic Strategy 15.25

FD-27 No Indices 15.91

FD-27 Full Gronbog Indices 20.40

REKO No Indices 24.14

HiLo No Indices 24.84

REKO QFIT Indices 36.92

HiLo Sweet 16 Indices 37.06



Unrestricted

Ace/5 No Indices 3.65

Thorp 10 No Indices 10.60

Speed No Indices 12.72

Speed Optimal Basic Strategy 15.28

FD-27 No Indices 15.92

FD-27 Full Gronbog Indices 20.42

REKO No Indices 24.17

HiLo No Indices 24.85

REKO QFIT Indices 37.04

HiLo Sweet 16 Indices 37.17





Heads Up c-SCORE 6.5/8 S17 DAS

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



Scenario System & c-SCORE

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

Play-All 1-8

Ace/5 No Indices 0.04

Thorp 10 No Indices 1.41

Speed No Indices 3.23

FD-27 No Indices 3.57

Speed Optimal Basic Strategy 4.06

FD-27 Full Gronbog Indices 4.67

REKO No Indices 5.55

HiLo No Indices 6.32

REKO QFIT Indices 9.96

HiLo Sweet 16 Indices 11.35



Play-All 1-10

Ace/5 No Indices 0.23

Thorp 10 No Indices 2.28

Speed No Indices 4.55

FD-27 No Indices 5.09

Speed Optimal Basic Strategy 5.66

FD-27 Full Gronbog Indices 6.33

REKO No Indices 7.70

HiLo No Indices 8.41

REKO QFIT Indices 13.10

HiLo Sweet 16 Indices 14.36



Play-All 1-12

Ace/5 No Indices 0.45

Thorp 10 No Indices 3.03

Speed No Indices 5.63

FD-27 No Indices 6.28

Speed Optimal Basic Strategy 6.94

FD-27 Full Gronbog Indices 7.59

REKO No Indices 9.42

HiLo No Indices 10.10

REKO QFIT Indices 15.62

HiLo Sweet 16 Indices 16.70



Back-Count 1-1

Ace/5 No Indices 2.97

Thorp 10 No Indices 8.24

Speed No Indices 12.03

FD-27 No Indices 13.68

Speed Optimal Basic Strategy 14.38

FD-27 Full Gronbog Indices 15.35

HiLo No Indices 19.06

REKO No Indices 19.31

HiLo Sweet 16 Indices 27.08

REKO QFIT Indices 28.26



Back-Count 1-2

Ace/5 No Indices 3.41

Thorp 10 No Indices 9.76

Speed No Indices 13.92

FD-27 No Indices 16.30

Speed Optimal Basic Strategy 16.72

FD-27 Full Gronbog Indices 18.48

REKO No Indices 22.51

HiLo No Indices 22.93

HiLo Sweet 16 Indices 32.99

REKO QFIT Indices 33.38



Back-Count 1-4

Ace/5 No Indices 3.64

Thorp 10 No Indices 10.45

Speed No Indices 14.61

FD-27 No Indices 17.09

Speed Optimal Basic Strategy 17.65

FD-27 Full Gronbog Indices 19.58

REKO No Indices 23.75

HiLo No Indices 24.21

HiLo Sweet 16 Indices 35.76

REKO QFIT Indices 35.80



Back-Count 1-8

Ace/5 No Indices 3.66

Thorp 10 No Indices 10.65

Speed No Indices 14.74

FD-27 No Indices 17.59

Speed Optimal Basic Strategy 17.87

FD-27 Full Gronbog Indices 19.68

REKO No Indices 24.13

HiLo No Indices 24.86

REKO QFIT Indices 36.61

HiLo Sweet 16 Indices 36.61



Back-Count 1-12

Ace/5 No Indices 3.67

Thorp 10 No Indices 10.68

Speed No Indices 14.77

FD-27 No Indices 17.75

Speed Optimal Basic Strategy 17.92

FD-27 Full Gronbog Indices 19.68

REKO No Indices 24.20

HiLo No Indices 25.04

REKO QFIT Indices 36.76

HiLo Sweet 16 Indices 37.08



Unrestricted

Ace/5 No Indices 3.68

Thorp 10 No Indices 10.69

Speed No Indices 14.78

FD-27 No Indices 17.78

Speed Optimal Basic Strategy 17.93

FD-27 Full Gronbog Indices 19.68

REKO No Indices 24.22

HiLo No Indices 25.05

REKO QFIT Indices 36.88

HiLo Sweet 16 Indices 37.19

 

[gronbog]

Let me clarify that these SCOREs (4 players) and c-SCORES (heads up) were for 6.5/8 S17 DAS SPL3. Also, as Peter's system (FD-27) was intended to be simple, we only compared its performance against other "simple" systems

 

[Peter JF]

I would also like to clarify that the results are what we have done are despite me not getting an answer to my original post at the top of this thread, with that information the system might perform better than the results above. The results above assumed a flat frequency distribution of the number of cards per hand being dealt in a heads-up situation.

It might make little difference, but we will never know without being able to test it. I would expect someone with good access to the data being collecting inside a simulation system could acquire this information quite easily, but it does not seem possible with the commercial simulation packages available. The link to Norm Wattenberger's statistics was the nearest I could find on the internet but not exactly what I was needing, as described in my first post.

Irrespective of this information, the basic approach of the system shows that, even without the originally requested information, it performs as the best of what can be considered the simplest but effective counting systems devised so far. Notably also being better than the Ten Count by Throp, which cannot be considered system! However, the Ten Count was the first recognised counting system that inspired the rest that have now followed with improved performance.

I accept that it may well be entry-level card counting, but I think that is a step that most people need from which they can decide whether to dedicate themselves to learning the more traditional counting systems, which in my opinion are quite demanding for Mr Average to learn.