Durango Bill's
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
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
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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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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