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Thread: I think I found a serious flaw in most Counting Systems

  1. #40


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    Quote Originally Posted by bigedge View Post
    In a double deck game, when penetration has reached 52 cards there are 107,632,809,909 possible [A,2,...,J,Q,K] permutations. This is the peak number possible in a double deck game.

    Of these, 95% of the most likely permutations are covered by the first 3,110,183,076, or about 2.89% of the total number of possible permutations.

    99% are covered by the first 8,071,684,634, or about 7.5%. This means that about 92.5% of all the possible deck compositions present at 52 card penetration are effectively useless at adding any value to analysis. Or, said another way, analysis that considers only the most likely 7.5% of all deck compositions will be quite sufficient in building an effective strategy.

    Of course, the number of permutations a 21 player needs to consider is significantly lower than 8,071,684,634, as the player is not concerned about the individual [T,J,Q,K] permutations but only their sum.
    I lose focus after 6,000,000.

    Think I'll stop sweating the small stuff starting precisely at 6:36:42 this evening

  2. #41


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    You may be comforted to know that 50% of the most likely deck compositions are covered by only 0.16% of all the possible permutations!

    The point of my post is to indicate that discrete analysis (as opposed to random simulation) is possible, once one realizes that the overwhelming majority of deck compositions can be safely ignored.

    Of course random simulation has its role in building a successful process for the player.

  3. #42
    Random number herder Norm's Avatar
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    Interesting. But, I don't think it's close to a proof of anything.
    "I don't think outside the box; I think of what I can do with the box." - Henri Matisse

  4. #43


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    Of course not. I'm only posting this as most players (I'm assuming) feel that discrete analysis is futile because of the overwhelming number of deck compositions to consider.

  5. #44


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    Below is a table indicating the number of blackjack-specific deck compositions that need to be analyzed to cover 99% of all possible permutations, for the penetration indicated. Permutations assume [T,J,Q,K] constitute a single rank.

    For example, in the case of 50 card penetration, exactly 624,655,486 deck compositions need to be analyzed to cover 99% of all possible deck compositions, or 8.21% of all possible deck compositions. Thus, 91.79% of all possible deck compositions can be ignored from analysis.

    Code:
     
                # of deck      
    Pen.      compositions
                covering 99%  % of all possible 
                of all perm.     permutations
    ----      ------------   -------------
    10            101,137        70.79%
    11            171,014        61.80%
    12            308,684        59.88%
    13            521,275        56.23%
    14            908,498        56.25%
    15          1,343,705        49.16%
    16          2,058,384        45.71%
    17          3,127,861        43.22%
    18          4,490,142        39.52%
    19          6,494,887        37.20%
    20          9,174,522        34.92%
    21         12,384,986        31.93%
    22         17,096,413        30.41%
    23         22,317,852        27.88%
    24         29,149,576        26.00%
    25         36,606,546        23.70%
    26         46,694,700        22.28%
    27         59,746,998        21.33%
    28         73,099,014        19.81%
    29         90,026,777        18.78%
    30        107,682,735        17.53%
    31        129,395,929        16.65%
    32        146,625,113        15.11%
    33        176,270,564        14.73%
    34        200,207,182        13.73%
    35        228,828,298        13.04%
    36        259,482,719        12.43%
    37        287,864,212        11.72%
    38        332,190,727        11.63%
    39        363,534,410        11.07%
    40        392,981,873        10.52%
    41        423,799,182        10.08%
    42        458,306,042         9.79%
    43        491,099,036         9.52%
    44        514,662,789         9.16%
    45        540,637,776         8.91%
    46        556,316,400         8.59%
    47        571,006,484         8.34%
    48        619,998,560         8.66%
    49        621,536,897         8.38%
    50        624,655,486         8.21%
    51        646,087,639         8.37%
    52        628,347,871         8.10%
    53        647,176,615         8.38%
    54        625,466,863         8.22%
    55        623,131,301         8.40%
    56        619,998,560         8.66%
    57        571,006,484         8.34%
    58        556,316,400         8.59%
    59        540,593,172         8.91%
    60        514,662,789         9.16%
    61        494,059,280         9.58%
    62        458,306,042         9.79%
    63        424,100,946        10.09%
    64        392,981,873        10.52%
    65        363,534,410        11.07%
    66        330,456,715        11.57%
    67        287,542,156        11.71%
    68        259,482,719        12.43%
    69        228,828,298        13.04%
    70        200,671,366        13.76%
    71        176,270,564        14.73%
    72        146,625,113        15.11%
    73        129,410,965        16.65%
    74        107,682,735        17.53%
    75         90,026,777        18.78%

  6. #45


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    Below is the same table as that above, but using 95% coverage of all possible deck compositions, instead of 99%.

    Code:
              # of deck      % of all possible
    Pen.      compositions   permutations
    ----      ------------   -------------
    10             73,260        51.27%
    11            127,351        46.02%
    12            214,436        41.60%
    13            360,838        38.92%
    14            575,564        35.64%
    15            962,534        35.22%
    16          1,428,820        31.73%
    17          2,005,023        27.71%
    18          2,651,519        23.34%
    19          3,730,072        21.37%
    20          5,462,365        20.79%
    21          7,276,702        18.76%
    22         10,187,029        18.12%
    23         12,326,254        15.40%
    24         16,471,616        14.69%
    25         20,412,042        13.22%
    26         23,370,956        11.15%
    27         30,116,120        10.75%
    28         36,657,451         9.93%
    29         44,474,380         9.28%
    30         53,659,991         8.73%
    31         62,716,736         8.07%
    32         71,043,145         7.32%
    33         79,171,997         6.61%
    34         96,275,464         6.60%
    35        110,625,160         6.30%
    36        119,178,451         5.71%
    37        130,274,670         5.30%
    38        139,848,468         4.90%
    39        153,749,660         4.68%
    40        158,534,803         4.24%
    41        181,117,805         4.31%
    42        196,481,153         4.20%
    43        214,096,262         4.15%
    44        216,436,226         3.85%
    45        221,238,290         3.65%
    46        225,362,201         3.48%
    47        241,134,207         3.52%
    48        260,540,118         3.64%
    49        237,450,011         3.20%
    50        250,870,421         3.30%
    51        250,061,942         3.24%
    52        251,486,534         3.24%
    53        250,106,546         3.24%
    54        250,870,421         3.30%
    55        236,167,331         3.18%
    56        260,540,118         3.64%
    57        241,928,007         3.53%
    58        226,156,001         3.49%
    59        221,238,290         3.65%
    60        218,624,090         3.89%
    61        214,096,262         4.15%
    62        193,037,573         4.12%
    63        181,117,805         4.31%
    64        158,534,803         4.24%
    65        153,749,660         4.68%
    66        139,668,162         4.89%
    67        130,274,670         5.30%
    68        119,199,970         5.71%
    69        110,625,160         6.30%
    70         96,275,464         6.60%
    71         79,171,997         6.61%
    72         71,043,145         7.32%
    73         63,049,586         8.11%
    74         53,659,991         8.73%
    75         44,474,380         9.28%

  7. #46


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    Hello,

    I've been away for a couple of weeks - looking into creating a freeware BJ simulation project. I'd like to discuss a few issues about this with the owner or administrator of this board. Can anyone pls tell me if that is Norm?

    Also, it seems to me that the person who posts under the ID of "Norm" is Norman Watternberger and that he is the person who is the owner and developer of the CVCX software. If that is correct, I'd like to ask if it might be possible for me to send him a private message?

    Thank you.

    I'm guessing that Norm would probably view a freeware project to create a BJ simulation as competition for his products. I'd like to ask if he might see any way that both may co-exist without necessarily seeing the other as an adversary?
    Last edited by Skyler62; 03-17-2017 at 03:38 AM.

  8. #47
    Random number herder Norm's Avatar
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    "I don't think outside the box; I think of what I can do with the box." - Henri Matisse

  9. #48


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    I apologize for the erroneous numbers I posted above. Below is a more concise table for a double deck showing correct values for 70%, 80%, 90%, 95%, and 99% coverage using the most likely deck compositions, specifically at the penetration indicated.

    Code:
                # of possible
                    deck              70%         80%         90%         95%         99%
      Pen.       compositions    coverage    coverage    coverage    coverage    coverage
     -----       ------------   ---------   ---------   ---------   ---------   ---------
        15          1,262,459        3.9%        5.9%        9.9%       14.4%       26.4%
        16          1,940,015        3.4%        5.2%        8.7%       12.8%       23.8%
        17          2,905,760        2.9%        4.5%        7.7%       11.4%       21.5%
        18          4,249,281        2.5%        4.0%        6.9%       10.3%       19.6%
        19          6,075,858        2.2%        3.5%        6.2%        9.3%       17.9%
        20          8,505,345        2.0%        3.1%        5.6%        8.5%       16.4%
        21         11,669,700        1.7%        2.8%        5.0%        7.7%       15.2%
        22         15,709,035        1.5%        2.5%        4.6%        7.1%       14.1%
        23         20,766,162        1.4%        2.3%        4.2%        6.5%       13.1%
        24         26,979,744        1.2%        2.1%        3.8%        6.0%       12.3%
        25         34,476,321        1.1%        1.9%        3.5%        5.6%       11.6%
        26         43,361,670        1.0%        1.7%        3.3%        5.2%       10.9%
        27         53,712,091        0.9%        1.6%        3.0%        4.9%       10.4%
        28         65,566,288        0.9%        1.5%        2.8%        4.6%       10.0%
        29         78,918,535        0.8%        1.4%        2.7%        4.4%        9.6%
        30         93,713,782        0.7%        1.3%        2.5%        4.2%        9.2%
        31        109,845,265        0.7%        1.2%        2.4%        4.0%        8.9%
        32        127,155,037        0.7%        1.2%        2.3%        3.8%        8.7%
        33        145,437,633        0.6%        1.1%        2.2%        3.7%        8.5%
        34        164,446,821        0.6%        1.1%        2.1%        3.6%        8.4%
        35        183,905,073        0.6%        1.0%        2.1%        3.5%        8.2%
        36        203,515,141        0.6%        1.0%        2.0%        3.4%        8.1%
        37        222,972,943        0.5%        1.0%        2.0%        3.4%        8.1%
        38        241,980,853        0.5%        0.9%        1.9%        3.3%        8.0%
        39        260,260,447        0.5%        0.9%        1.9%        3.3%        8.0%
        40        277,563,784        0.5%        0.9%        1.9%        3.3%        8.0%
        41        293,682,397        0.5%        0.9%        1.9%        3.3%        8.1%
        42        308,453,343        0.5%        0.9%        1.9%        3.2%        8.1%
        43        321,761,913        0.5%        0.9%        1.9%        3.2%        8.2%
        44        333,540,933        0.5%        0.9%        1.8%        3.3%        8.2%
        45        343,766,869        0.5%        0.9%        1.8%        3.3%        8.3%
        46        352,453,183        0.5%        0.9%        1.8%        3.3%        8.4%
        47        359,641,573        0.5%        0.9%        1.8%        3.3%        8.4%
        48        365,391,868        0.5%        0.9%        1.9%        3.3%        8.5%
        49        369,771,439        0.5%        0.9%        1.9%        3.3%        8.5%
        50        372,845,029        0.5%        0.9%        1.9%        3.3%        8.6%
        51        374,665,863        0.5%        0.9%        1.9%        3.3%        8.6%
        52        375,268,773        0.5%        0.9%        1.9%        3.3%        8.6%
        53        374,665,863        0.5%        0.9%        1.9%        3.3%        8.6%
        54        372,845,029        0.5%        0.9%        1.9%        3.3%        8.6%
        55        369,771,439        0.5%        0.9%        1.9%        3.3%        8.5%
        56        365,391,868        0.5%        0.9%        1.9%        3.3%        8.5%
        57        359,641,573        0.5%        0.9%        1.8%        3.3%        8.4%
        58        352,453,183        0.5%        0.9%        1.8%        3.3%        8.4%
        59        343,766,869        0.5%        0.9%        1.8%        3.3%        8.3%
        60        333,540,933        0.5%        0.9%        1.8%        3.3%        8.2%
        61        321,761,913        0.5%        0.9%        1.9%        3.2%        8.2%
        62        308,453,343        0.5%        0.9%        1.9%        3.2%        8.1%
        63        293,682,397        0.5%        0.9%        1.9%        3.3%        8.1%
        64        277,563,784        0.5%        0.9%        1.9%        3.3%        8.0%
        65        260,260,447        0.5%        0.9%        1.9%        3.3%        8.0%
        66        241,980,853        0.5%        0.9%        1.9%        3.3%        8.0%
        67        222,972,943        0.5%        1.0%        2.0%        3.4%        8.1%
        68        203,515,141        0.6%        1.0%        2.0%        3.4%        8.1%
        69        183,905,073        0.6%        1.0%        2.1%        3.5%        8.2%
        70        164,446,821        0.6%        1.1%        2.1%        3.6%        8.4%
        71        145,437,633        0.6%        1.1%        2.2%        3.7%        8.5%
        72        127,155,037        0.7%        1.2%        2.3%        3.8%        8.7%
        73        109,845,265        0.7%        1.2%        2.4%        4.0%        8.9%
        74         93,713,782        0.7%        1.3%        2.5%        4.2%        9.2%
        75         78,918,535        0.8%        1.4%        2.7%        4.4%        9.6%

  10. #49
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    Can you please re-address your original
    thesis re: permutations of possible hands ?

    To me, not only have you noted nothing
    of value, but it is getting "long in the tooth."


  11. #50


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    That there is an alternative to random simulation in developing a theoretical strategy of blackjack. That is, by analyzing a small portion of the complete set of permutations possible, a result virtually devoid of uncertainty can be developed.

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