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.