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Bridge Probabilities and Combinatorics



Bridge Probabilities and Statistics - Point Count

Computer Program by Bill Butler

What is the probability that you will be dealt a hand worth "N" points?


   One of the best methods of evaluating the strength of a hand is by adding up the point count. This method awards 4 points for every Ace in a hand, 3 points for each King, 2 points for each Queen, and 1 point for each Jack. Additional points are awarded for distributional strength such as a short suit, but here we will just give the probabilities for raw honor count power.

   The highest possible honor count that can exist in a hand would have all four Aces, four Kings, four Queens, and one of the four Jacks for a total of 37 points. At the other end of the scale there are over 2 billion hands that have a zero point count (10 high or worse). Of these, COMBIN(32, 13) = 347,373,600 are Yarboroughs (9 high or less). We also note there are COMBIN(52, 13) = 635,013,599,600 different hands that could be dealt.

   For each row in the table below, column 1 gives the point count via the 4, 3, 2, 1 analysis. Note that there are many combinations that can produce a given count. For example a hand that has 2 Aces and a Queen, or another hand that has 4 Queens and 2 Jacks would both be included in the "10" row.

   The second column shows the exact number of possible hands that will produce the given point count (Honors only - distribution is not counted). For the third column we divide the total combinations in the second column by the total number of all hands (635,013,559,600) to get the probability of being dealt this particular point count. The fourth column gives the cumulative probability of receiving a particular point count or higher. Finally the fifth column shows the average number of honor cards that a hand will have, given that the hand has a particular honor count.

   High point count hands have a very low probability, and hence scientific notation is used. For example, to express the probability of getting a 37 point hand as a fixed point decimal number, you have to move the decimal point 12 places further to the left (e.g. .00000000000629908)


Bridge Honor Count Combinations for 13 cards (1 hand)

The total number of possible 13 card hands is: COMBIN(52,13) = 635,013,559,600


Honor             Total    Honor Count     Cumulative       Avg. Nbr.
Count             Hands    Probability     Probability      of Honors
---------------------------------------------------------------------
  37                  4    6.29908e-012    6.29908e-012       13.0000
  36                 60    9.44862e-011    1.00785e-010       12.4000
  35                624    9.82656e-010    1.08344e-009       12.0769
  34              4,484    7.06127e-009    8.14471e-009       11.4585
  33             22,360    3.52118e-008    4.33566e-008       11.2161
  32            109,156    1.71896e-007    2.15252e-007       10.6851
  31            388,196    6.11319e-007    8.26571e-007       10.4401
  30          1,396,068    2.19849e-006    3.02506e-006       10.0376
  29          4,236,588    6.67165e-006    9.69671e-006        9.7116
  28         11,790,760    1.85677e-005    2.82644e-005        9.4187
  27         31,157,940    4.90666e-005    7.7331e-005         9.0614
  26         74,095,248    0.000116683     0.000194014         8.7857
  25        167,819,892    0.000264278     0.000458292         8.4670
  24        354,993,864    0.000559034     0.00101733          8.1655
  23        710,603,628    0.00111904      0.00213636          7.8697
  22      1,333,800,036    0.00210043      0.00423679          7.5769
  21      2,399,507,844    0.00377867      0.00801546          7.2797
  20      4,086,538,404    0.00643536      0.0144508           6.9817
  19      6,579,838,440    0.0103617       0.0248125           6.7023
  18     10,192,504,020    0.0160508       0.0408634           6.3982
  17     14,997,082,848    0.0236169       0.0644803           6.1113
  16     21,024,781,756    0.0331092       0.0975895           5.8196
  15     28,090,962,724    0.0442368       0.14183             5.5275
  14     36,153,374,224    0.0569332       0.19876             5.2273
  13     43,906,944,752    0.0691433       0.2679              4.9381
  12     50,971,682,080    0.0802687       0.34817             4.6450
  11     56,799,933,520    0.0894468       0.43762             4.3279
  10     59,723,754,816    0.0940511       0.53167             4.0415
   9     59,413,313,872    0.0935623       0.62523             3.7356
   8     56,466,608,128    0.0889219       0.71415             3.4192
   7     50,979,441,968    0.0802809       0.79443             3.0811
   6     41,619,399,184    0.065541        0.85998             2.8059
   5     32,933,031,040    0.0518619       0.91184             2.4620
   4     24,419,055,136    0.0384544       0.95029             2.0525
   3     15,636,342,960    0.0246236       0.97492             1.7448
   2      8,611,542,576    0.0135612       0.98848             1.4186
   1      5,006,710,800    0.00788442      0.99636             1.0000
   0      2,310,789,600    0.00363896      1                   0.0000



   Similar calculations can be made for the combined point count totals if you add the point count total in your hand to the point count total in your partnerís hand. The table below shows the number of ways any possible point count total could occur and the probability of each of these possibilities.

Bridge Honor Count Combinations for 26 cards (2 hands)

The total number of possible combined hands is: COMBIN(52, 26) = 495,918,532,948,104

Honor                  Total    Honor Count      Cumulative      Avg. Nbr.
Count                  Hands    Probability      Probability     of Honors
--------------------------------------------------------------------------
  40             254,186,856    5.12558e-007     0.00000051       16.0000
  39           2,403,221,184    4.846e-006       0.00000536       15.0000
  38           9,913,287,384    1.99897e-005     0.00002535       14.2424
  37          31,673,222,784    6.38678e-005     0.00008922       13.7840
  36          89,195,378,184    0.000179859      0.00026908       13.3201
  35         211,712,342,400    0.00042691       0.00069598       12.9023
  34         459,808,617,240    0.000927186      0.00162317       12.5337
  33         920,662,591,680    0.00185648       0.00347965       12.1553
  32       1,691,764,828,380    0.00341138       0.00689103       11.8072
  31       2,914,543,903,680    0.00587706       0.01276809       11.4616
  30       4,734,398,485,800    0.00954673       0.02231481       11.1296
  29       7,257,585,574,080    0.0146346        0.03694945       10.8019
  28      10,533,038,026,200    0.0212395        0.05818890       10.4784
  27      14,596,737,921,600    0.0294337        0.08762264       10.1614
  26      19,258,439,527,560    0.0388339        0.12645652        9.8462
  25      24,259,718,677,440    0.0489188        0.17537528        9.5360
  24      29,295,317,098,380    0.0590728        0.23444812        9.2246
  23      33,876,647,618,880    0.0683109        0.30275903        8.9182
  22      37,522,340,994,600    0.0756623        0.37842134        8.6110
  21      39,905,485,171,200    0.0804678        0.45888916        8.3055
  20      40,775,251,597,080    0.0822217        0.54111084        8.0000
  19      39,905,485,171,200    0.0804678        0.62157866        7.6945
  18      37,522,340,994,600    0.0756623        0.69724097        7.3890
  17      33,876,647,618,880    0.0683109        0.76555188        7.0818
  16      29,295,317,098,380    0.0590728        0.82462472        6.7754
  15      24,259,718,677,440    0.0489188        0.87354348        6.4640
  14      19,258,439,527,560    0.0388339        0.91237736        6.1538
  13      14,596,737,921,600    0.0294337        0.94181110        5.8386
  12      10,533,038,026,200    0.0212395        0.96305055        5.5216
  11       7,257,585,574,080    0.0146346        0.97768519        5.1981
  10       4,734,398,485,800    0.00954673       0.98723191        4.8704
   9       2,914,543,903,680    0.00587706       0.99310897        4.5384
   8       1,691,764,828,380    0.00341138       0.99652035        4.1928
   7         920,662,591,680    0.00185648       0.99837683        3.8447
   6         459,808,617,240    0.000927186      0.99930402        3.4663
   5         211,712,342,400    0.00042691       0.99973092        3.0977
   4          89,195,378,184    0.000179859      0.99991078        2.6799
   3          31,673,222,784    6.38678e-005     0.99997465        2.2160
   2           9,913,287,384    1.99897e-005     0.99999464        1.7576
   1           2,403,221,184    4.846e-006       0.99999949        1.0000
   0             254,186,856    5.12558e-007     1.00000000        0.0000

   The symmetry that exists above vs. below the 20-point line is not just a coincidence. For example the numbers for a 26 point hand are the same as for a 14 point hand. (If your side has only 14 high card points, then the opponents have 26 high card points. Itís a 50-50 toss-up as to which side gets the good hands.



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