Mathematical Tables

Mathematical Probabilities Percentage

The numerous combinations of 52 cards and their probabilites and/or percentages of distribution and/or pattern and/or occurence have been studied by many mathematicians, arithmeticians, probability theorists, and mathematical statisticians. The source of the tables below are from The Official Encyclopedia of Bridge, published by the American Contract Bridge League in the year 1984.

Note to all readers: Some or several of theses percentages and probabilities may have changed as a result of newer studies. Many universities and colleges and other educational organizations continue to employ the 52 cards used in the game of bridge as the foundation of their doctorate or academic degree. Such studies may have resulted in more accurate formulae and results than those presented on this web page. Any reference to updated and possibly more accurate results will be greatly appreciated.

The following mathematical tables may/can/are used to determine the percentages of various distribution patterns, both for hand patterns and suit patterns. The numbers are expressed in percentage of hands. The percentage expectation of a particular pattern with the suits identified is expressed in the last column.

Pattern Total Specific
4-4-3-2 21.5512 1.796
4-3-3-3 10.5361 2.634
4-4-4-1 2.9932 0.748
5-3-3-2 15.5168 1.293
5-4-3-2 12.9307 0.539
5-4-2-2 10.5797 0.882
5-5-2-1 3.1739 0.264
5-4-4-0 1.2433 0.104
5-5-3-0 0.8952 0.075
6-3-2-2 5.6425 0.470
6-4-2-1 4.7021 0.196
6-3-3-1 3.4482 0.287
6-4-3-0 1.3262 0.055
6-5-1-1 0.7053 0.059
6-5-2-0 0.6511 0.027
6-6-1-0 0.0723 0.006
7-3-2-1 1.8808 0.078
7-2-2-2 0.5129 0.128
7-4-1-1 0.3918 0.033
7-4-2-0 0.3617 0.015
7-3-3-0 0.2652 0.022
7-5-1-0 0.1085 0.005
7-6-0-0 0.0056 0.0005
8-2-2-1 0.1924 0.016
8-3-1-1 0.1176 0.010
8-3-2-0 0.1085 0.005
8-4-1-0 0.052 0.002
8-5-0-0 0.0031 0.0003
9-2-1-1 0.0178 0.001
9-3-1-0 0.0100 0.0004
9-2-2-0 0.0082 0.0007
9-4-0-0 0.0010 0.00008
10-2-1-0 0.0011 0.00004
10-1-1-1 0.0004 0.0001
10-3-0-0 0.00015 0.00001
11-1-1-0 0.00002 0.000002
11-2-0-0 0.00001 0.000001
12-1-0-0 0.0000003 0.00000003
13-0-0-0 0.0000000006 0.0000000002

Probable Percentage Frequency of Distribution Patterns

The following table presents the expectation of holding specific point counts, using the 4-3-2-1 count.

Probable Frequency of High Card Content

Point Count Percentage Point Count Percentage
0 .3639 16 3.3109
1 .7884 17 2.3617
2 1.3561 18 1.6051
3 2.4624 19 1.0362
4 3.8454 20 .6435
5 5.1862 21 .3779
6 6.5541 22 .2100
7 8.0281 23 .1119
8 8.8922 24 .0559
9 9.3562 25 .0264
10 9.4051 26 .0117
11 8.9447 27 .0049
12 8.0269 28 .0019
13 6.9143 29 .0007
14 5.6933 30 .0002
15 4.4237 31-37 .0001

 

The following table presents the probability, even before dealing the cards, of holding an exact number of cards in a specified suit. It must be noted that the number of times the specified number of cards can be expected in any suit during the course of 100 deals is four times as great.

Number of Cards   Percentage
0 1.279
1 8.006
2 20.587
3 28.633
4 23.861
5 12.469
6 4.156
7 0.882
8 0.117
9 0.009
10 0.0004
11 0.000009
12 0.00000008
13 0.00000000016

Probability of Holding an Exact Number of Cards in a Specified Suit

The following table present the probability of distribution of the remaining cards in a suit for:
A. a one-hand holding in column (1)
B. among the other three hands in column (2)
C. and expressed as a percentage in column (3)

Probability of Distribution of Cards in Three Hidden Hands

(1) (2) (3)   (1) (2) (3)
0 6-4-3 25.921 4 3-3-3 11.039
5-4-4 24-301 4-4-1 9.408
5-5-3 17.497 6-2-1 4.927
6-5-2 12.725 5-4-0 2.605
7-4-2 7.069 6-3-0 1.390
7-3-3 5.184 5 3-3-2 31.110
8-3-2 2.121 4-3-1 25.925
7-5-1 2.121 4-2-2 21.212
6-6-1 1.414 5-2-1 12.727
8.4.1 0.884 5-3-0 3.590
1 5-4-3 40.377 4-4-0 2.493
6-4-2 14.683 6-1-1 1.414
6-3-3 10.767 6-2-0 1.305
5-5-2 9.911 6 3-2-2 33.939
4-4-4 9.347 4-2-1 28.282
7-3-2 5.873 3-3-1 20.740
6-5-1 4.405 4-3-0 7.977
7-4-1 2.447 5-1-1 4.242
8-3-1 0.734 5-2-0 3.916
8-2-2 0.601 6-1-0 0.870
2 4-4-3 26.170 7 3-2-1 53.333
5-4-2 25.695 2-2-2 14.545
5-3-3 18.843 4-1-1 11.111
6-3-2 13.704 4-2-0 10.256
6-4-1 5.710 3-3-0 7.521
5-5-1 3.854 5-1-0 3.077
7-3-1 2.284 8 2-2-1 41.211
7-2-2 1.869 3-1-1 25.185
6-5-0 0.791 3-2-0 23.247
3 4-3-3 27.598 4.1.0 9.686
5-3-2 27.096 5-0-0 0.671
4-4-2 18.817 9 2-1-1 48.080
5-4-1 11.290 3-1-0 27.122
6-3-1 6.021 2-2-0 22.191
6-2-2 4.927 4-0-0 2.608
7-2-1 1.642 10 2-1-0 66.572
6-4-0 1.158 1-1-1 24.040
5-5-0 0.782 3-0-0 9.388
4 4-3-2 45.160 11 1-1-0 68.421
5-3-1 13.548 2-0-0 31.579
5-2-2 11.085

The following table presents the probability of distribution of cards in two given hands.

A. (1) shows the number of cards in the two known hands.

B. (2) shows the number of outstanding cards in the two hidden hands.

C. (3) shows the ways in which these cards may be divided.

D. (4) shows the percentage of cases in which the distribution in column (3) occurs.

E. (5) shows the number of cases applicable.

F. (6) is the result of dividing the percentage (4) by (5), and indicates the probability
that one opponent will hold particular specified cards.

 

(1) (2) (3) (4) (5) (6)
11 2 1-1 52.00 2 26.0000
2-0 48.00 2 26.0000
10 3 2-1 78.00 6 13.0000
3-0 22.00 2 11.000
9 4 3-1 49.74 8 6.2175
2-2 40.70 6 6.7833
4-0 9.57 2 4.7850
8 5 3-2 67.83 20 3.392
4-1 28.26 10 2.826
5-0 3.91 2 1.9550
7 6 4-2 48.45 30 1.6150
3-3 35.53 20 1.7765
5-1 14.53 12 1.2108
6-0 1.49 2 .7450
6 7 4-3 62.17 70 1.0362
5-2 30.52 42 7.2667
6-1 6.78 14 .4843
7-0 0.52 2 .2600
5 8 5-3 47.12 112 .4207
4-4 32.72 70 .4674
6-2 17.14 56 .3061
7-1 2.86 16 .1788
8-0 0.16 2 .0800
4 9 5-4 58.90 252 .2337
6-3 31.41 168 .1870
7-2 8.57 72 .1190
8-1 1.07 18 .0595
9-0 0.05 2 .0250
3 10 6-4 46.20 420 .1100
5-5 31.18 252 .1237
7-3 18.48 240 .0770
8-2 3.78 90 .0420
9-1 0.35 20 .0175
10-0 0.01 2 .0050
2 11 6-5 57.17 924 .0619
7-4 31.76 660 .0481
8-3 9.53 330 .0289
9-2 1.44 110 .0131
10-1 0.10 22 .0400
11-0 0.002 2 .0010
1 12 7-5 45.74 1584 .02889
6-6 30.49 924 .0330
8-4 19.06 990 .0193
9-3 4.23 440 .0096
10-2 0.46 132 .0034
11-1 .02 24 .0008
12-0 0.0003 2 .0002
0 13 7-6 56.62 3432 .0165
8-5 31.85 2574 .0124
9-4 9.83 1430 .0061
10.3 1.57 572 .0028
11-2 0.12 156 .0007
12-1 0.003 26 .0001
13-0 0.00002 2 .00001

Probability of Distribution of Cards in Two Hidden Hands

A residue is said to be favorably divided when it is divided as evenly as possible. In the following table:
A. column (1) shows the number of cards outstanding in each of the two suits in the two hidden hands.
B. column (2) shows the percentage of cases in which both residues will divide as evenly as possible.
C. column (3) shows the percentage of cases in which at least one residue will divide favorably.

 

Probability of Distribution of Two Residues Between Two Hidden Hands

(1) (2) (3)
8-8 11.87 53.57
8-7 21.77 73.13
8-6 12.44 55.81
8.5 23.10 77.45
8-4 13.86 59.56
7-7 40.42 83.93
7-6 23.10 74.60
7-5 43.31 86.69
7-4 25.99 76.88
6-6 13.20 57.86
6-5 24.75 78.61
6-4 14.85 61.37
5-5 46.75 88.90
5-4 28.05 80.47
5-3 53.29 92.53

 

Odds and Odds

The odds in the game of bridge has been a fascinating subject for many bridge players throughout the years. Mathematicians have devoted much time to finding formulas for calculating these odds. After their calculations, we present perhaps just a sampling of the different possibilities in the constellation of the cards.

The longest possible bridge auction in the game of bridge is presented on this web page.

The number of possible deals: 53,644,737,765,488,792,839,237,440,000.

The possible number of bridge auctions, as has been mathematically calculated, is: 128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557.

The number of possible different hands that a named player can receive: 635,013,559,600.

The number of possible auctions by North, if East/West passes: 68,719,476,735.

The number of possible auctions by North, if East/West do not pass: 128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557.

The odds against each player having a complete suit: 2,235,197,406,895,366,368,301,559,999 to 1.

The odds against one player holding a Yarborough: 1,827 to 1. The odds against two players holding a Yarborough: 546,000,000 to 1.