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 |
6.78 |
2 |
3.3900 |
|
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.
The longest possible auction is presented below, solely for entertainment.
|
North |
East |
South |
West |
|
|
|
Pass |
Pass |
Pass |
1 |
1 |
|
|
Pass |
Pass |
Double |
Pass |
2 |
|
|
Pass |
Redouble |
Pass |
Pass |
3 |
|
|
1 |
Pass |
Pass |
Double |
4 |
|
|
Pass |
Pass |
Redouble |
Pass |
5 |
|
|
Pass |
1 |
Pass |
Pass |
6 |
|
|
Double |
Pass |
Pass |
Redouble |
7 |
|
|
Pass |
Pass |
1 |
Pass |
8 |
|
|
Pass |
Double |
Pass |
Pass |
9 |
|
|
Redouble |
Pass |
Pass |
1 NT |
10 |
|
|
Pass |
Pass |
Double |
Pass |
11 |
|
|
Pass |
Redouble |
Pass |
Pass |
12 |
|
|
2 |
Pass |
Pass |
Double |
13 |
|
|
Pass |
Pass |
Redouble |
Pass |
14 |
|
|
Pass |
2 |
Pass |
Pass |
15 |
|
|
Double |
Pass |
Pass |
Redouble |
16 |
|
|
Pass |
Pass |
2 |
Pass |
17 |
|
|
Pass |
Double |
Pass |
Pass |
18 |
|
|
Redouble |
Pass |
Pass |
2 |
19 |
|
|
Pass |
Pass |
Double |
Pass |
20 |
|
|
Pass |
Redouble |
Pass |
Pass |
21 |
|
|
2 NT |
Pass |
Pass |
Double |
22 |
|
|
Pass |
Pass |
Redouble |
Pass |
23 |
|
|
Pass |
3 |
Pass |
Pass |
24 |
|
|
Double |
Pass |
Pass |
Redouble |
25 |
|
|
Pass |
Pass |
3 |
Pass |
26 |
|
|
Pass |
Double |
Pass |
Pass |
27 |
|
|
Redouble |
Pass |
Pass |
3 |
28 |
|
|
Pass |
Pass |
Double |
Pass |
29 |
|
|
Pass |
Redouble |
Pass |
Pass |
30 |
|
|
3 |
Pass |
Pass |
Double |
31 |
|
|
Pass |
Pass |
Redouble |
Pass |
32 |
|
|
Pass |
3 NT |
Pass |
Pass |
33 |
|
|
Double |
Pass |
Pass |
Redouble |
34 |
|
|
Pass |
Pass |
4 |
Pass |
35 |
|
|
Pass |
Double |
Pass |
Pass |
36 |
|
|
Redouble |
Pass |
Pass |
4 |
37 |
|
|
Pass |
Pass |
Double |
Pass |
38 |
|
|
Pass |
Redouble |
Pass |
Pass |
39 |
|
|
4 |
Pass |
Pass |
Double |
40 |
|
|
Pass |
Pass |
Redouble |
Pass |
41 |
|
|
Pass |
4 |
Pass |
Pass |
42 |
|
|
Double |
Pass |
Pass |
Redouble |
43 |
|
|
Pass |
Pass |
4 NT |
Pass |
44 |
|
|
Pass |
Double |
Pass |
Pass |
45 |
|
|
Redouble |
Pass |
Pass |
5 |
46 |
|
|
Pass |
Pass |
Double |
Pass |
47 |
|
|
Pass |
Redouble |
Pass |
Pass |
48 |
|
|
5 |
Pass |
Pass |
Double |
49 |
|
|
Pass |
Pass |
Redouble |
Pass |
50 |
|
|
Pass |
5 |
Pass |
Pass |
51 |
|
|
Double |
Pass |
