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Thread: Zenfighter: EoR's HITTING 17 - 12 Table 1

  1. #14
    Cacarulo
    Guest

    Cacarulo: Re: First practical application

    Good work ZF! I could check "some" of your EoRs if you want but you'll have to tell me which ones would you prefer. Maybe the ones that you consider more crucial or more different from the ones in ToB.
    Obviously, I don't have the time to check them all

    Just let me know.

    Sincerely,
    Cac

  2. #15
    Zenfighter
    Guest

    Zenfighter: Re: First practical application

    Cac!

    You don't have to worry; I'll need you for sure for the next coming tables, for double checks. You guessed it! Hard doubling vs ace or ten (American rules), not to mention for splitting and/ or resplitting, during the next comming summer. The idea is to keep Don's blood pressure at reasonable levels. :-)

    For the "piece of cake", just relax. They are fine. Comparing them with Griffin's entries
    the only noticeable are slightly discrepancies with the ss, a natural consequence of the single precision digits vs the rounding ones. Nothing to write home, anyway.

    Btw, enjoyed your insurance tables. Note that with the EoR's you gave me I've got:

    Insure if TC => 1.38992

    If you give me data and full deck favorabilities for two, four and six decks, I'll let you
    know what the EoR's derived indexes are.

    Appreciated answer.

    Sincerely

    Z


  3. #16
    Cacarulo
    Guest

    Cacarulo: Re: First practical application

    > You don't have to worry; I'll need you for
    > sure for the next coming tables, for double
    > checks. You guessed it! Hard doubling vs ace
    > or ten (American rules), not to mention for
    > splitting and/ or resplitting, during the
    > next comming summer. The idea is to keep
    > Don's blood pressure at reasonable levels.
    > :-)

    Ok. The splitting part is not going to be easy though.

    > For the "piece of cake", just
    > relax. They are fine. Comparing them with
    > Griffin's entries
    > the only noticeable are slightly
    > discrepancies with the ss, a natural
    > consequence of the single precision digits
    > vs the rounding ones. Nothing to write home,
    > anyway.

    Ok.

    > Btw, enjoyed your insurance tables. Note
    > that with the EoR's you gave me I've got:
    > Insure if TC => 1.38992

    That's pretty good but I still think Snyder's formula is not that accurate. Not to mention trying to get indices for unbalanced systems in TC mode. However and in this particular case it seems that SF is doing a better job than Pete's formula.

    > If you give me data and full deck
    > favorabilities for two, four and six decks,
    > I'll let you
    > know what the EoR's derived indexes are.

    No problem. It would be interesting to see what you get.

    Sincerely,
    Cac

  4. #17
    Cacarulo
    Guest

    Cacarulo: Complete set of Insurance EoRs

    ZenFighter,

    Here you have two sets of Insurance EoRs for any number of decks. The first set is the "traditional" that uses the full pack. The second set adjusts for the ace removed.
    The indices generated by using any of these sets are the same since insurance is a linear play.
    For other plays such as 16vT -which is not linear- a set adjusted for the removal of a ten would
    be more accurate than the traditional.

    1) Full Pack

         +--------------------+--------------------+--------------------+--------------------+ 
    | 1D | 2D | 3D | 4D |
    +----+--------------------+--------------------+--------------------+--------------------+
    | A | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 2 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 3 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 4 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 5 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 6 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 7 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 8 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | 9 | 1.80995475113122 | 0.89619118745332 | 0.59553349875931 | 0.44593088071349 |
    | T | -4.07239819004525 | -2.01643017176998 | -1.33995037220844 | -1.00334448160535 |
    | m | -7.69230769230769 | -7.69230769230769 | -7.69230769230769 | -7.69230769230769 |
    | ss | 95.82113388341763 | 23.49239035071819 | 10.37380933322661 | 5.81648974843682 |
    | ck | 0.00000000000000 | 0.00000000000000 | 0.00000000000001 | 0.00000000000001 |
    +----+--------------------+--------------------+--------------------+--------------------+
    | 5D | 6D | 7D | 8D |
    +----+--------------------+--------------------+--------------------+--------------------+
    | A | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 2 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 3 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 4 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 5 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 6 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 7 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 8 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | 9 | 0.35640035640036 | 0.29680930002473 | 0.25429116338207 | 0.22242817423540 |
    | T | -0.80190080190080 | -0.66782092505565 | -0.57215511760966 | -0.50046339202966 |
    | m | -7.69230769230769 | -7.69230769230769 | -7.69230769230769 | -7.69230769230769 |
    | ss | 3.71537051073731 | 2.57680099699930 | 1.89142187639558 | 1.44712306129057 |
    | ck | 0.00000000000000 | 0.00000000000001 | 0.00000000000001 | 0.00000000000001 |
    +----+--------------------+--------------------+--------------------+--------------------+


    2) Full Pack - Ace

         +--------------------+--------------------+--------------------+--------------------+ 
    | 1D | 2D | 3D | 4D |
    +----+--------------------+--------------------+--------------------+--------------------+
    | A | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 2 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 3 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 4 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 5 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 6 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 7 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 8 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | 9 | 1.88235294117647 | 0.91376356367790 | 0.60326770004189 | 0.45026030673983 |
    | T | -4.11764705882353 | -2.02741290691034 | -1.34478424801005 | -1.00605037287182 |
    | m | -5.88235294117647 | -6.79611650485437 | -7.09677419354839 | -7.24637681159420 |
    | ss | 100.76124567474051 | 24.08348855724444 | 10.54644370454471 | 5.88879914330485 |
    | ck | -0.00000000000000 | -0.00000000000000 | 0.00000000000000 | 0.00000000000000 |
    +----+--------------------+--------------------+--------------------+--------------------+
    | 5D | 6D | 7D | 8D |
    +----+--------------------+--------------------+--------------------+--------------------+
    | A | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 2 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 3 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 4 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 5 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 6 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 7 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 8 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | 9 | 0.35916314986082 | 0.29872419873457 | 0.25569608693667 | 0.22350270647809 |
    | T | -0.80362754781359 | -0.66901773674930 | -0.57303319483129 | -0.50113497468133 |
    | m | -7.33590733590734 | -7.39549839228296 | -7.43801652892562 | -7.46987951807229 |
    | ss | 3.75223421803959 | 2.59807323554546 | 1.90479049284131 | 1.45606530097737 |
    | ck | 0.00000000000000 | -0.00000000000000 | 0.00000000000000 | 0.00000000000000 |
    +----+--------------------+--------------------+--------------------+--------------------+


    In particular I use the first set (1D) for calculating algebraic insurance indices from 1D to 8D.
    Please, let me know what indices do you get with these EoRs.

    Sincerely,
    Cac

  5. #18
    Zenfighter
    Guest

    Zenfighter: Re: Comparisons

     

    Decks EoR?s derived

    1 1.39093
    2 2.33007
    3 2.64313
    4 2.79965
    5 2.89357
    6 2.95618
    7 3.0009
    8 3.03444



    It?s worth remarking, that all indexes derived by algebraic approximations using Pete?s formula
    are more precise, in the sense that they approach closer to the true figures. The only exception seems to be the single deck one.

    So it seems to me, that Arnold?s formula is still a practical tool under these circumstances:

    a) Aimed basically at one level balanced counts.
    b) Will give you fair indexes if you limit yourself to single deck exclusively.

    As a standard tool Moss formula is proven to be superior. The data is conclusive.

    Sincerely

    Z

  6. #19
    alienated
    Guest

    alienated: Re: First practical application

    Thanks for posting these, Zenfighter. I've had a lot of fun in the past utilizing the EOR tables in ToB, and greatly appreciate your contributions in this area.

  7. #20
    Zenfighter
    Guest

    Zenfighter: Re: Glad to see you round here

    Coming from the author of the best descriptive EoR?s articles I?ve read in my life, I can?t resist the temptation to link them here (they can be accessed for free on the Web, anyway), in order that DD?s subscribers will appreciate the beautiful inferences that can be extracted from any given table.

    Alienated articles:
    1. www.cardcounter.com/best.pl?read=7
    2. www.cardcounter.com/best.pl?read=6


    Priceless, gotta believe!

    Sincerely

    Z


  8. #21
    Zenfighter
    Guest

    Zenfighter: Re: EoR's Table 5 (missing line added)

    DEALER 8


    HITTING 17 ? 12


     
    17 16 15 14

    A -2.2913 -1.05088 -0.626284 -0.457921
    2 -2.33871 -1.38594 -0.314613 0.0424502
    3 -2.85997 -2.43771 -1.34779 -0.255674
    4 -2.47536 -2.19894 -1.76435 -0.7419
    5 1.37724 -2.36073 -2.04525 -1.60912
    6 1.6552 1.72456 -2.03732 -1.77118
    7 1.83993 1.92535 1.90629 -1.89921
    8 2.18038 2.37604 2.31794 2.28689
    9 1.78107 0.201884 0.325984 0.448319
    T 0.282875 0.801591 0.896347 0.989336

    m -12.3434 5.22865 9.02769 12.8089

    ss 41.2946 34.31359 26.08121 19.5066

    Cks -0.00002 -0.000002 -0.000005 -0.0000018


     
    13 12

    A -0.291204 -0.12559
    2 0.152055 0.261938
    3 0.120518 0.24809
    4 0.282184 0.563732
    5 -0.608089 0.394101
    6 -1.38626 -0.387063
    7 -1.62222 -1.22888
    8 -1.53287 -1.30068
    9 0.567841 -3.09399
    T 1.07951 1.16709

    m 16.5563 20.2719

    ss 12.45866 18.99195

    Cks -0.000005 0.000018





    The fact that hard 17 is the most volatile of the stiff hitting situations is revealed by the 12th column figure of 41.0.
    A player who split three eights and drew (8,9), (8,7,9), and (8,9) would be more than 5% better off to hit the last total of 17 even though the hand was dealt from a full pack!


    P. Griffin

  9. #22
    Zenfighter
    Guest

    Zenfighter: Re: Further practical applications

    Griffin?s examples revisited

    Another use is to find some of the ?composition? dependent departures from the simplified basic strategy defined in Chapter Two.

    Before removing any number of cards from a specific table we need an adjusting factor (actually a multiplier) to reflect exactly the increased effect the removed cards will have upon the obviously smaller number of the remaining ones.

    Adjusting factor = 51/ (52 ? nr) where nr = number of removed cards

    1) Should you hit or stand with (4, 4, 4, 4) v 8?

    From table 5 (the corrected one!) we have:

    ((4 * -2.19894) + 2.37604) * 51/47 = -6.966079

    Adjusting m we have then:

    m = 5.22865 ? 6.966079 = -1. 737429 that?s

    You are better off standing, something that makes perfect sense with the pack ripped off from 4s.

    2) Just for drill the reader might confirm the 2.3% advantage hitting (6, 4, 6) v T mentioned in Chapter One

    From table 3 we have:

    ((2 * 1.64458) + (-1.72785) + 1.11513)) * 51/48 = 2.843718

    Adjusting m again we have:

    m = -0.445861 + 2.843718 = 2.397857 that?s

    You should hit the hand. A clearly demonstration that 3-card composition dependent hands totalling 16 vs a dealer?s T should be hit if one or more sixes are included in your hand. Statements coming out from a pocket calculator!

    Stay tuned.

    Sincerely

    Z

  10. #23
    Zenfighter
    Guest

    Zenfighter: Re: Cac, did you check The Waterkooler? *NM*


  11. #24
    Cacarulo
    Guest

    Cacarulo: Re: Cac, did you check The Waterkooler?

    E-mail sent! Sorry, I've forgotten about the existence of those pages. Will have to drop there more often.

    Sincerely,
    Cac

  12. #25
    Zenfighter
    Guest

    Zenfighter: Re: EV's inferences

    Approximating EV?s with a pocket calculator.

    As Griffin taught us, these methods with the aid of EoR?s tables are only approximate. You won?t be able to match perfectly exact expectations derived with the straightforward computations of powerful BJ?s algorithms. Let?s work an example with the aid of Cacarulo?s EV tables from bjmath.com.

    		standing	hitting 

    8,7 vs T -.51402 -.47479



    From the above figures we learn that the player is better by a 3.923% when hitting.

    See if we can approach this exact figure with the aid of Table 3 (hitting 15) and a pocket calculator.

    Remove the 3 cards:

    (0.0852692 + (-0.536984) + 1.19918) * 51/49 = 0.777974

    Adjusting m:

    m = 3.11027 + 0.777974 = 3.888244

    That is, our prediction is a roughly 3.9% advantage for hitting over standing, for this particular hand.

    Extracting algebraic indexes is another feature of these tables. I?ll post an example next time.

    Sincerely

    Z

  13. #26
    Zenfighter
    Guest

    Zenfighter: Re: Algebraic indexes

    Extracting EoR?s derived algebraic indexes is another important feature of the above tables.
    Following Snyder?s methodology outlined in his technical paper (Algebraic Approximation of Optimum Blackjack Strategy) and with the aid of our newly revised hit-stand tables, I?ve got the following generic indexes rounded to the nearest decimal.


    SD, s17

     
    2 3 4 5 6 7 8 9 T A

    16 4.1 -0.1
    15 3.6

    13 -0.0 -1.4
    12 4.4 2.6 0.6 -1.0 -0.1*

    * -2.5 (hit soft 17)



    Sincerely

    Z

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