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Mathematics of Bridge

Table 1-Probabilities of High Card Point Counts for One Hand

HCP	Percentage	HCP	Percentage
0	.364	19	1.036
1	.789	20	.644
2	1.356	21	.378
3	2.462	22	.210
4	3.845	23	.112
5	5.186	24	.056
6	6.554	25	.026
7	8.028	26	.012
8	8.892	27	.0049
9	9.356	28	.0019
10	9.405	29	.00067
11	9.945	30	.00022
12	8.027	31	.00006
13	6.914	32	.00002
14	5.693	33	.000004
15	4.424	34	.0000007
16	3.311	35	.0000001
17	2.362	36	.000000009
18	1.605	37	.0000000006

Table 2-Probabilities of High Card Point Counts for a Partnership

HCP	Percentage	HCP	Percentage
0	.00005	21	8.047
1	.0005	22	7.566
2	.002	23	6.831
3	.006	24	5.907
4	.018	25	4.892
5	.043	26	3.883
6	.093	27	2.943
7	.196	28	2.124
8	.341	29	1.463
9	.588	30	.955
10	.955	31	.588
11	1.463	32	.341
12	2.124	33	.186
13	2.942	34	.093
14	3.983	35	.043
15	4.892	36	.018
16	5.907	37	.006
17	6.931	38	.002
18	7.566	39	.0005
19	8.047	40	.00005
20	8.222

Probability that either partnership will have enough to bid game (>= 26 pts.) = 25.29% (about 1 in every 3.95 deals)
Probability that either partnership will have enough to bid slam (>= 33 pts.) = .697% (about 1 in every 143.5 deals)
Probability that either partnership will have enough to bid grand slam (>= 37 pts.) = .0171% (about 1 in every 5848 deals)

Table 3-Probable Percentage Frequency of Distribution Patterns

Pattern	Total	Pattern	Total
4-4-3-2	21.5512	7-4-2-0	0.3617
4-3-3-3	10.5361	7-3-3-0	0.2652
4-4-4-1	2.9932	7-5-1-0	0.1085
5-3-3-2	15.5168	7-6-0-0	0.0056
5-4-3-1	12.9307	8-2-2-1	0.1924
5-4-2-2	10.5797	8-3-1-1	0.1176
5-5-2-1	3.1739	8-3-2-0	0.1085
5-4-4-0	1.2433	8-4-1-0	0.0452
5-5-3-0	0.8952	8-5-0-0	0.0031
6-3-2-2	5.6425	9-2-1-1	0.0178
6-4-2-1	4.7021	9-3-1-0	0.0100
6-3-3-1	3.4482	9-2-2-0	0.0082
6-4-3-0	1.3262	9-4-0-0	0.0010
6-5-1-1	0.7053	10-2-1-0	0.0011
6-5-2-0	0.6511	10-1-1-1	0.0004
6-6-1-0	0.0723	10-3-0-0	0.00015
7-3-2-1	1.8808	11-1-1-0	0.00002
7-2-2-2	0.5129	11-2-0-0	0.00001
7-4-1-1	0.3918	12-1-0-0	0.00001
		13-0-0-0	0.0000000006

Table 4-Probability of Holding an Exact Number of Cards of a Specified Suit

Number of Cards	%
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

Number of different hands a named player can receive = 52C13 = 635,013,559,600.
Number of different hands a second player can receive = 39C13 = 8,122,425,444.
Number of different hands the 3rd and 4th players can receive = 26C13 = 10,400,600.
Number of possible deals = 52!/(13!)^4 = 53,644,737,765,488,792,839,237,440,000.
Number of possible auctions with North as dealer, assuming that East and West pass throughout = 2^36 - 1 = 68,719,476,735.
Number of possible auctions with North as dealer, assuming that East and West do not pass throughout = 128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557.
Odds against each player having a complete suit = 2,235,197,406,895,366,368,301,559,999 to 1.
Odds against receiving a hand with 4 aces, 4 kings, 4 queens, and 1 jack = 158,753,389,899 to 1.
Odds against receiving a perfect hand (a hand that will take 13 tricks in no-trump no matter what) = 169,066,442 to 1.
Odds against a yarborough = 1827 to 1.
Odds against both members of a partnership receiving yarboroughs = 546,000,000 to 1.
Odds against a hand with no card higher than 10 = 274 to 1.
Odds against a hand with no card higher than jack = 52 to 1.
Odds against a hand with no card higher than queen = 11 to 1.
Odds against a hand with no aces = slightly higher than 2 to 1.
Odds against being dealt four aces = 378 to 1.
Odds against being dealt four honors in one suit = 22 to 1.
Odds against being dealt five honors in one suit = 500 to 1.
Odds against being dealt at least one singleton = slightly higher than 2 to 1.
Odds against having at least one void = 19 to 1.
Odds that two partners will be dealt 26 named cards between them (for example, all the spades & diamonds) = 495,918,532,918,103 to 1 against.
Odds that no players wil be dealt a singleton or void = 4 to 1 against.
Data obtained from The Official Encyclopedia of Bridge, Newly Revised, Fifth Edition. Copyright 1994 by American Contract Bridge League, Inc. Main Page
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