# Mathematics of Bridge

Table 1-Probabilities of High Card Point Counts for One Hand
HCP Percentage HCP Percentage
0.364191.036
1.78920.644
21.35621.378
32.46222.210
43.84523.112
55.18624.056
66.55425.026
78.02826.012
88.89227.0049
99.35628.0019
109.40529.00067
119.94530.00022
128.02731.00006
136.91432.00002
145.69333.000004
154.42434.0000007
163.31135.0000001
172.36236.000000009
181.60537.0000000006

Table 2-Probabilities of High Card Point Counts for a Partnership
HCP Percentage HCP Percentage
0.00005218.047
1.0005227.566
2.002236.831
3.006245.907
4.018254.892
5.043263.883
6.093272.943
7.196282.124
8.341291.463
9.58830.955
10.95531.588
111.46332.341
122.12433.186
132.94234.093
143.98335.043
154.89236.018
165.90737.006
176.93138.002
187.56639.0005
198.04740.00005
208.222

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
PatternTotalPatternTotal
4-4-3-221.55127-4-2-00.3617
4-3-3-310.53617-3-3-00.2652
4-4-4-12.99327-5-1-00.1085
5-3-3-215.51687-6-0-00.0056
5-4-3-112.93078-2-2-10.1924
5-4-2-210.57978-3-1-10.1176
5-5-2-13.17398-3-2-00.1085
5-4-4-01.24338-4-1-00.0452
5-5-3-00.89528-5-0-00.0031
6-3-2-25.64259-2-1-10.0178
6-4-2-14.70219-3-1-00.0100
6-3-3-13.44829-2-2-00.0082
6-4-3-01.32629-4-0-00.0010
6-5-1-10.705310-2-1-00.0011
6-5-2-00.651110-1-1-10.0004
6-6-1-00.072310-3-0-00.00015
7-3-2-11.880811-1-1-00.00002
7-2-2-20.512911-2-0-00.00001
7-4-1-10.391812-1-0-00.00001
13-0-0-00.0000000006

Table 4-Probability of Holding an Exact Number of Cards of a Specified Suit
Number of Cards%
01.279
18.006
220.587
328.633
423.861
512.469
64.156
70.882
80.117
90.009
100.0004
110.000009
120.00000008
130.00000000016
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.