MATHEMATICAL TABLES

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.

Probable Percentage Frequency of Distribution Patterns

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

 

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.

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

 

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

 

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.

Probability of Distribution of Cards in Two Hidden Hands

(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

 

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

 

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 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.

 

 

 

 

 

 

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