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 first column in the table below lists all possible
suit length combinations (Each number is a suit length). The second
column lists the total number of possible hands that exist for each
suit distribution. These must add to the total combinations that exist
for a Bridge hand which is: COMBIN( 52, 13) = 635,013,559,600. The
probability of a given suit distribution is simply the Total Hands for
this distribution divided by the total possible hands (635,013,559,600).
Distribution
Total Hands Probability
---------------------------------------------------
13, 0, 0,
0
4 6.29908 E-12
12, 1, 0,
0
2,028 3.19363 E-09
11, 2, 0,
0
73,008 1.14971 E-07
11, 1, 1,
0
158,184 2.49103 E-07
10, 3, 0,
0
981,552 1.54572 E-06
10, 2, 1,
0
6,960,096 1.09605 E-05
10, 1, 1,
1
2,513,368 3.95798 E-06
9, 4, 0,
0
6,134,700 9.66074 E-06
9, 3, 1,
0
63,800,880 0.000100472
9, 2, 2,
0
52,200,720 8.22041 E-05
9, 2, 1,
1
113,101,560 0.000178109
8, 5, 0,
0
19,876,428 3.13008 E-05
8, 4, 1,
0
287,103,960 0.000452123
8, 3, 2,
0
689,049,504 0.00108509
8, 3, 1,
1
746,470,296 0.00117552
8, 2, 2,
1
1,221,496,848 0.00192358
7, 6, 0,
0
35,335,872 5.56459 E-05
7, 5, 1,
0
689,049,504 0.00108509
7, 4, 2,
0
2,296,831,680 0.00361698
7, 4, 1,
1
2,488,234,320 0.0039184
7, 3, 3,
0
1,684,343,232 0.00265245
7, 3, 2,
1
11,943,524,736 0.0188083
7, 2, 2,
2
3,257,324,928 0.00512954
6, 6, 1,
0
459,366,336 0.000723396
6, 5, 2,
0
4,134,297,024 0.00651056
6, 5, 1,
1
4,478,821,776 0.00705311
6, 4, 3,
0
8,421,716,160 0.0132623
6, 4, 2,
1
29,858,811,840 0.0470207
6, 3, 3,
1
21,896,462,016 0.0344819
6, 3, 2,
2
35,830,574,208 0.0564249
5, 5, 3,
0
5,684,658,408 0.00895203
5, 5, 2,
1
20,154,697,992 0.0317390
5, 4, 4,
0
7,895,358,900 0.0124334
5, 4, 3,
1
82,111,732,560 0.129307
5, 4, 2,
2
67,182,326,640 0.105797
5, 3, 3,
2
98,534,079,072 0.155168
4, 4, 4,
1
19,007,345,500 0.0299322
4, 4, 3,
2
136,852,887,600 0.215512
4, 3, 3,
3
66,905,856,160 0.105361
Total
635,013,559,600 1.000000
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