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Bridge Probabilities and Combinatorics



Bridge Probabilities and Statistics - Suit Distribution

What is the probability of getting a particular suit distribution?


Computer Program by Bill Butler


   Everyone is used to getting hands where the maximum number of cards in any one suit is 5 or less. Occasionally six, seven, or even longer suits show up. The table below shows the number of combinations and probability of getting all possible suit distributions. We assume the deck is completely randomized before the hands are dealt. In practice there are usually only 3 or 4 shuffles between hands in ordinary social bridge. This is not enough to completely randomize the cards when the input to the shuffle is tricks consisting of clumps of cards in the same suit. As a result, long suited "weird" hands show up more frequently than would otherwise be expected.

   The table below lists all possible suit length distributions (Each number is a suit length). The “Total Hands” column shows how many different bridge hands are possible for the given distribution. For example you could have all 13 cards in any one of the 4 suits.

   The “Probability” column shows the probability of getting a bridge hand with the indicated distribution. This probability is equal to the Total Hands for the distribution divided by the total possible hands. The total number of hands must sum to COMBIN(52, 13) = 635,013,559,600.

                                      Nbr. of 13
Count      Distribution               Card Hands         Probability
---------------------------------------------------------------------
  1)      13,  0,  0,  0                       4         6.29908E-012
  2)      12,  1,  0,  0                   2,028         3.19363E-009
  3)      11,  2,  0,  0                  73,008         1.14971E-007
  4)      11,  1,  1,  0                 158,184         2.49103E-007
  5)      10,  3,  0,  0                 981,552         1.54572E-006
  6)      10,  2,  1,  0               6,960,096         1.09605E-005
  7)      10,  1,  1,  1               2,513,368         3.95798E-006
  8)       9,  4,  0,  0               6,134,700         9.66074E-006
  9)       9,  3,  1,  0              63,800,880         0.000100472
 10)       9,  2,  2,  0              52,200,720         8.22041E-005
 11)       9,  2,  1,  1             113,101,560         0.000178109
 12)       8,  5,  0,  0              19,876,428         3.13008E-005
 13)       8,  4,  1,  0             287,103,960         0.000452123
 14)       8,  3,  2,  0             689,049,504         0.00108509
 15)       8,  3,  1,  1             746,470,296         0.00117552
 16)       8,  2,  2,  1           1,221,496,848         0.00192358
 17)       7,  6,  0,  0              35,335,872         5.56459E-005
 18)       7,  5,  1,  0             689,049,504         0.00108509
 19)       7,  4,  2,  0           2,296,831,680         0.00361698
 20)       7,  4,  1,  1           2,488,234,320         0.0039184
 21)       7,  3,  3,  0           1,684,343,232         0.00265245
 22)       7,  3,  2,  1          11,943,524,736         0.0188083
 23)       7,  2,  2,  2           3,257,324,928         0.00512954
 24)       6,  6,  1,  0             459,366,336         0.000723396
 25)       6,  5,  2,  0           4,134,297,024         0.00651056
 26)       6,  5,  1,  1           4,478,821,776         0.00705311
 27)       6,  4,  3,  0           8,421,716,160         0.0132623
 28)       6,  4,  2,  1          29,858,811,840         0.0470207
 29)       6,  3,  3,  1          21,896,462,016         0.0344819
 30)       6,  3,  2,  2          35,830,574,208         0.0564249
 31)       5,  5,  3,  0           5,684,658,408         0.00895203
 32)       5,  5,  2,  1          20,154,697,992         0.031739
 33)       5,  4,  4,  0           7,895,358,900         0.0124334
 34)       5,  4,  3,  1          82,111,732,560         0.12931
 35)       5,  4,  2,  2          67,182,326,640         0.1058
 36)       5,  3,  3,  2          98,534,079,072         0.15517
 37)       4,  4,  4,  1          19,007,345,500         0.0299322
 38)       4,  4,  3,  2         136,852,887,600         0.21551
 39)       4,  3,  3,  3          66,905,856,160         0.10536
Totals                           635,013,559,600         1.0000000



   There has been some interest in what the distribution probabilities would be if you combined 2 bridge hands. For example, if you combined your hand with your partner’s hand, then you would have a giant hand of 26 cards. The question arises as to the possible suit distributions that might exist in this giant 26-card hand. Mathematically, this would be the same as randomly picking 26 cards from a deck of cards.

    The table below lists all possible suit length distributions (Each number is a suit length) for one of these giant 26-card hands. The “Nbr. Of 26 Card Hands” column shows how many different hands are possible for the given distribution. For example you could have all 13 cards in both of two suits. There are 6 possible suit combinations. (Spades/Hearts, Spades/Diamonds, Spades/Clubs, Hearts/Diamonds, Hearts/Clubs, Diamonds/Clubs).

   The “Probability” column shows the probability of getting a 26 card hand with the indicated distribution. This probability is equal to the Total Hands for the distribution divided by the total possible hands. The total number of hands must sum to COMBIN(52, 26) = 495,918,532,948,104.


                                        Nbr. of 26
Count        Distribution               Card Hands         Probability
-----------------------------------------------------------------------
  1)        13, 13,  0,  0                       6         1.20988E-014
  2)        13, 12,  1,  0                   4,056         8.17876E-012
  3)        13, 11,  2,  0                 146,016         2.94435E-010
  4)        13, 11,  1,  1                 158,184         3.18972E-010
  5)        13, 10,  3,  0               1,963,104         3.95852E-009
  6)        13, 10,  2,  1               6,960,096         1.40348E-008
  7)        13,  9,  4,  0              12,269,400         2.47408E-008
  8)        13,  9,  3,  1              63,800,880         1.28652E-007
  9)        13,  9,  2,  2              52,200,720         1.05261E-007
 10)        13,  8,  5,  0              39,752,856         8.01601E-008
 11)        13,  8,  4,  1             287,103,960         5.78934E-007
 12)        13,  8,  3,  2             689,049,504         1.38944E-006
 13)        13,  7,  6,  0              70,671,744         1.42507E-007
 14)        13,  7,  5,  1             689,049,504         1.38944E-006
 15)        13,  7,  4,  2           2,296,831,680         4.63147E-006
 16)        13,  7,  3,  3           1,684,343,232         3.39641E-006
 17)        13,  6,  6,  1             459,366,336         9.26294E-007
 18)        13,  6,  5,  2           4,134,297,024         8.33665E-006
 19)        13,  6,  4,  3           8,421,716,160         1.69821E-005
 20)        13,  5,  5,  3           5,684,658,408         1.14629E-005
 21)        13,  5,  4,  4           7,895,358,900         1.59207E-005
 22)        12, 12,  2,  0                 158,184         3.18972E-010
 23)        12, 12,  1,  1                 171,366         3.45553E-010
 24)        12, 11,  3,  0               6,960,096         1.40348E-008
 25)        12, 11,  2,  1              24,676,704         4.97596E-008
 26)        12, 10,  4,  0              63,800,880         1.28652E-007
 27)        12, 10,  3,  1             331,764,576         6.6899E-007
 28)        12, 10,  2,  2             271,443,744         5.47356E-007
 29)        12,  9,  5,  0             287,103,960         5.78934E-007
 30)        12,  9,  4,  1           2,073,528,600         4.18119E-006
 31)        12,  9,  3,  2           4,976,468,640         1.00349E-005
 32)        12,  8,  6,  0             689,049,504         1.38944E-006
 33)        12,  8,  5,  1           6,718,232,664         1.3547E-005
 34)        12,  8,  4,  2          22,394,108,880         4.51568E-005
 35)        12,  8,  3,  3          16,422,346,512         3.3115E-005
 36)        12,  7,  7,  0             459,366,336         9.26294E-007
 37)        12,  7,  6,  1          11,943,524,736         2.40836E-005
 38)        12,  7,  5,  2          53,745,861,312         0.000108376
 39)        12,  7,  4,  3         109,482,310,080         0.000220767
 40)        12,  6,  6,  2          35,830,574,208         7.22509E-005
 41)        12,  6,  5,  3         197,068,158,144         0.00039738
 42)        12,  6,  4,  4         136,852,887,600         0.000275958
 43)        12,  5,  5,  4         184,751,398,260         0.000372544
 44)        11, 11,  4,  0              52,200,720         1.05261E-007
 45)        11, 11,  3,  1             271,443,744         5.47356E-007
 46)        11, 11,  2,  2             222,090,336         4.47836E-007
 47)        11, 10,  5,  0             689,049,504         1.38944E-006
 48)        11, 10,  4,  1           4,976,468,640         1.00349E-005
 49)        11, 10,  3,  2          11,943,524,736         2.40836E-005
 50)        11,  9,  6,  0           2,296,831,680         4.63147E-006
 51)        11,  9,  5,  1          22,394,108,880         4.51568E-005
 52)        11,  9,  4,  2          74,647,029,600         0.000150523
 53)        11,  9,  3,  3          54,741,155,040         0.000110383
 54)        11,  8,  7,  0           4,134,297,024         8.33665E-006
 55)        11,  8,  6,  1          53,745,861,312         0.000108376
 56)        11,  8,  5,  2         241,856,375,904         0.000487694
 57)        11,  8,  4,  3         492,670,395,360         0.00099345
 58)        11,  7,  7,  1          35,830,574,208         7.22509E-005
 59)        11,  7,  6,  2         429,966,890,496         0.000867011
 60)        11,  7,  5,  3       1,182,408,948,864         0.00238428
 61)        11,  7,  4,  4         821,117,325,600         0.00165575
 62)        11,  6,  6,  3         788,272,632,576         0.00158952
 63)        11,  6,  5,  4       2,956,022,372,160         0.0059607
 64)        11,  5,  5,  5         665,105,033,736         0.00134116
 65)        10, 10,  6,  0           1,684,343,232         3.39641E-006
 66)        10, 10,  5,  1          16,422,346,512         3.3115E-005
 67)        10, 10,  4,  2          54,741,155,040         0.000110383
 68)        10, 10,  3,  3          40,143,513,696         8.09478E-005
 69)        10,  9,  7,  0           8,421,716,160         1.69821E-005
 70)        10,  9,  6,  1         109,482,310,080         0.000220767
 71)        10,  9,  5,  2         492,670,395,360         0.00099345
 72)        10,  9,  4,  3       1,003,587,842,400         0.00202369
 73)        10,  8,  8,  0           5,684,658,408         1.14629E-005
 74)        10,  8,  7,  1         197,068,158,144         0.00039738
 75)        10,  8,  6,  2       1,182,408,948,864         0.00238428
 76)        10,  8,  5,  3       3,251,624,609,376         0.00655677
 77)        10,  8,  4,  4       2,258,072,645,400         0.00455331
 78)        10,  7,  7,  2         788,272,632,576         0.00158952
 79)        10,  7,  6,  3       5,780,665,972,224         0.0116565
 80)        10,  7,  5,  4      10,838,748,697,920         0.0218559
 81)        10,  6,  6,  4       7,225,832,465,280         0.0145706
 82)        10,  6,  5,  5       9,754,873,828,128         0.0196703
 83)         9,  9,  8,  0           7,895,358,900         1.59207E-005
 84)         9,  9,  7,  1         136,852,887,600         0.000275958
 85)         9,  9,  6,  2         821,117,325,600         0.00165575
 86)         9,  9,  5,  3       2,258,072,645,400         0.00455331
 87)         9,  9,  4,  4       1,568,106,003,750         0.00316202
 88)         9,  8,  8,  1         184,751,398,260         0.000372544
 89)         9,  8,  7,  2       2,956,022,372,160         0.0059607
 90)         9,  8,  6,  3      10,838,748,697,920         0.0218559
 91)         9,  8,  5,  4      20,322,653,808,600         0.0409798
 92)         9,  7,  7,  3       7,225,832,465,280         0.0145706
 93)         9,  7,  6,  4      36,129,162,326,400         0.072853
 94)         9,  7,  5,  5      24,387,184,570,320         0.0491758
 95)         9,  6,  6,  5      32,516,246,093,760         0.0655677
 96)         8,  8,  8,  2         665,105,033,736         0.00134116
 97)         8,  8,  7,  3       9,754,873,828,128         0.0196703
 98)         8,  8,  6,  4      24,387,184,570,320         0.0491758
 99)         8,  8,  5,  5      16,461,349,584,966         0.0331937
100)         8,  7,  7,  4      32,516,246,093,760         0.0655677
101)         8,  7,  6,  5     117,058,485,937,536         0.23604
102)         8,  6,  6,  6      26,012,996,875,008         0.0524542
103)         7,  7,  7,  5      26,012,996,875,008         0.0524542
104)         7,  7,  6,  6      52,025,993,750,016         0.10491
Totals                         495,918,532,948,104         1.0000000



 
Partnership Distributional Fit

   For any of the standard 13 card bridge hands, it might also be of interest to see what the probabilities are that your partner might have a given number of cards in your longest suit. For example, if you have a 5 card suit that you would like to use a "trump" suit, what is the probability that your partner has:
4 cards in this suit
or
3 cards in this suit
or
2 cards in this suit
Etc.

   The following tables show the probabilities that if you have "N" cards in your longest suit, then your partner might have "K" cards in the same suit. Note that if you have two or more suits with the same longest length, then the tables refer only to the single longest suit of interest. Also, the calculations assume that you have no information about any of the other hands. (Note: A "Pass" by an opponent does reveal some information.)

   In all cases, after your 13 cards have been removed from the deck, then there are Combin(39, 13) = 8,122,425,445 possible hands that your partner could have.

If your longest suit has 13 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      0            8,122,425,444        1.000000



If your longest suit has 12 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      1            2,707,475,148        0.333333
      0            5,414,950,296        0.666667



If your longest suit has 11 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      2              854,992,152        0.105263
      1            3,704,965,992        0.456140
      0            3,562,467,300        0.438596



If your longest suit has 10 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      3              254,186,856        0.031294
      2            1,802,415,888        0.221906
      1            3,755,033,100        0.462304
      0            2,310,789,600        0.284495



If your longest suit has 9 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      4               70,607,460        0.008693
      3              734,317,584        0.090406
      2            2,503,355,400        0.308203
      1            3,337,807,200        0.410937
      0            1,476,337,800        0.181761



If your longest suit has 8 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      5               18,156,204        0.002235
      4              262,256,280        0.032288
      3            1,311,281,400        0.161440
      2            2,860,977,600        0.352232
      1            2,741,770,200        0.337556
      0              927,983,760        0.114250




If your longest suit has 7 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      6                4,272,048        0.000526
      5               83,304,936        0.010256
      4              578,506,500        0.071223
      3            1,851,220,800        0.227915
      2            2,903,050,800        0.357412
      1            2,128,903,920        0.262102
      0              573,166,440        0.070566



If your longest suit has 6 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      7                  906,192        0.000112
      6               23,560,992        0.002901
      5              220,884,300        0.027194
      4              981,708,000        0.120864
      3            2,257,928,400        0.277987
      2            2,709,514,080        0.333584
      1            1,580,549,880        0.194591
      0              347,373,600        0.042767



If your longest suit has 5 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      8                  169,911        0.000021
      7                5,890,248        0.000725
      6               73,628,100        0.009065
      5              441,768,600        0.054389
      4            1,411,205,250        0.173742
      3            2,483,721,240        0.305786
      2            2,370,824,820        0.291886
      1            1,128,964,200        0.138993
      0              206,253,075        0.025393



If your longest suit has 4 cards, then the possible combinations for your partner are:

Nbr. Cards           Number  of       Probability
Partner has           Possible          of this
In this suit        Combinations      Combination
      9                   27,405        0.000003
      8                1,282,554        0.000158
      7               21,375,900        0.002632
      6              171,007,200        0.021054
      5              737,468,550        0.090794
      4            1,802,700,900        0.221941
      3            2,523,781,260        0.310718
      2            1,966,582,800        0.242118
      1              778,439,025        0.095838
      0              119,759,850        0.014744




Partner-Partner Suit Distributional Combinations

   At the top of the page, we posted a table of the 39 possible suit distributions that might exist in a bridge hand. For each of these 39 distributions, we might be interested in the suit distributional fit (or misfit) that might exist in your partner’s hand, and the probability of each of these suit fits (or misfits).

   For example, let’s assume that you have a 5, 4, 3, 1 suit distribution. We ask:
How many possible suit distributions are possible in your partner’s hand, and what is the probability of each of these? The distribution in your partner’s hand might be:

      Hand 1
   Distribution
   5,  4,  3,  1

         Hand 2        Hand 2          Hand  2        Cumulative
         Count      Distribution     Probability     Probability
            1)     8,  5,  0,  0      0.00000002      0.00000002
            2)     8,  4,  1,  0      0.00000016      0.00000017
            3)     8,  4,  0,  1      0.00000019      0.00000036
            4)     8,  3,  2,  0      0.00000047      0.00000082
            5)     8,  3,  1,  1      0.00000124      0.00000206
            Etc.


   We note that the 5, 4, 3, 1 distribution in your hand can be in any arbitrary suits; but if you are going to match your hand with your partner’s hand, then the suits in your partner’s hand must be matched with the suits in your hand. Thus we define a 5, 4, 3, 1 distribution in your hand as:

Suit 1: 5 cards
Suit 2: 4 cards
Suit 3: 3 cards
Suit 4: 1 card

Then we match whatever these 4 suits are with the suits that are in your partner’s hand. Thus the “1)  8, 5, 0, 0” above becomes:

Your partner has 8 cards in your “Suit 1”
Your partner has 5 cards in your “Suit 2”
Your partner has 0 cards in your “Suit 3”
Your partner has 0 cards in your “Suit 4”

Similarly for “2)  8, 4, 1, 0”:

Your partner has 8 cards in your “Suit 1”
Your partner has 4 cards in your “Suit 2”
Your partner has 1 card in your “Suit 3”
Your partner has 0 cards in your “Suit 4”

   In all, if you have a 5, 4, 3, 1 suit distribution, there are 494 possible suit distributions that your partner might have. If we consider all 39 of the possible suit distributions that you might have (See table at the top of the page), then there are 16,393 possible combined suit fits (or misfits) that could occur between your hand and your partner’s hand. Here is a file that lists all of them. http://www.durangobill.com/BridgePics/BrSuitComb2Hands.txt

   For example, if you have a 5, 4, 3, 1 suit distribution and want to see all 494 possible suit combinations that might exist in your partner’s hand, scroll down the file list until you see:

Hand 1        Hand 1
Count      Distribution
 34)       5,  4,  3,  1

         Hand 2        Hand 2          Hand  2        Cumulative
         Count      Distribution     Probability     Probability
            1)     8,  5,  0,  0      0.00000002      0.00000002
            2)     8,  4,  1,  0      0.00000016      0.00000017
            3)     8,  4,  0,  1      0.00000019      0.00000036
            4)     8,  3,  2,  0      0.00000047      0.00000082
            5)     8,  3,  1,  1      0.00000124      0.00000206
            Etc.

for all 494 suit distributions that your partner might have.

   Each row also gives the probability that your partner has the particular suit distributional fit as well as the cumulative probability which is a running sum of all the Hand 2 probabilities.

   There are some entries in the file table that show 0.00000000 for the probability of a given combination. These entries actually do have a small probability of occurring, but this probability is so low that it doesn’t show up in the first 8 digits to the right of the decimal point.






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