# Bridge probabilities

In the game of bridge mathematical probabilities play a significant role. Different declarer play strategies lead to success depending on the distribution of opponent's cards. To decide which strategy has highest likelihood of success, the declarer needs to have at least an elementary knowledge of probabilities.

The tables below specify the various prior probabilities, i.e. the probabilities in the absence of any further information. During bidding and play, more information about the hands becomes available, allowing players to improve their probability estimates.

## Probability of suit distributions in two hidden hands

This table[1] represents the different ways that two to thirteen particular cards may be distributed, or may lie or split, between two unknown 13-card hands (before the bidding and play, or a priori).

The table also shows the number of combinations of particular cards that match any numerical split and the probabilities for each combination.

These probabilities follow directly from the law of Vacant Places.

Number
of Cards
Distribution Probability Combinations Individual
Probability
2 1 - 1 0.52 2 0.26
2 - 0 0.48 2 0.24
3 2 - 1 0.78 6 0.13
3 - 0 0.22 2 0.11
4 2 - 2 0.41 6 0.0678~
3 - 1 0.50 8 0.0622~
4 - 0 0.10 2 0.0478~
5 3 - 2 0.68 20 0.0339~
4 - 1 0.28 10 0.02826~
5 - 0 0.04 2 0.01956~
6 3 - 3 0.36 20 0.01776~
4 - 2 0.48 30 0.01615~
5 - 1 0.15 12 0.01211~
6 - 0 0.01 2 0.00745~
7 4 - 3 0.62 70 0.00888~
5 - 2 0.31 42 0.00727~
6 - 1 0.07 14 0.00484~
7 - 0 0.01 2 0.00261~
8 4 - 4 0.33 70 0.00467~
5 - 3 0.47 112 0.00421~
6 - 2 0.17 56 0.00306~
7 - 1 0.03 16 0.00178~
8 - 0 0.00 2 0.00082~

## Probability of HCP distribution

High card points (HCP) are usually counted using the Milton Work scale of 4/3/2/1 points for each Ace/King/Queen/Jack respectively. The a priori probabilities that a given hand contains no more than a specified number of HCP is given in the table below.[1] To find the likelihood of a certain point range, one simply subtracts the two relevant cumulative probabilities. So, the likelihood of being dealt a 12-19 HCP hand (ranges inclusive) is the probability of having at most 19 HCP minus the probability of having at most 11 hcp, or: 0.986 − 0.652 = 0.334.[2]

HCP Probability HCP Probability HCP Probability HCP Probability HCP Probability
0 0.0036 8 0.3748 16 0.9355 24 0.9995 32 1.0000
1 0.0115 9 0.4683 17 0.9591 25 0.9998 33 1.0000
2 0.0251 10 0.5624 18 0.9752 26 0.9999 34 1.0000
3 0.0497 11 0.6518 19 0.9855 27 1.0000 35 1.0000
4 0.0882 12 0.7321 20 0.9920 28 1.0000 36 1.0000
5 0.1400 13 0.8012 21 0.9958 29 1.0000 37 1.0000
6 0.2056 14 0.8582 22 0.9979 30 1.0000
7 0.2858 15 0.9024 23 0.9990 31 1.0000

## Hand pattern probabilities

A hand pattern denotes the distribution of the thirteen cards in a hand over the four suits. In total 39 hand patterns are possible, but only 13 of them have an a priori probability exceeding 1%. The most likely pattern is the 4-4-3-2 pattern consisting of two four-card suits, a three-card suit and a doubleton.

Note that the hand pattern leaves unspecified which particular suits contain the indicated lengths. For a 4-4-3-2 pattern, one needs to specify which suit contains the three-card and which suit contains the doubleton in order to identify the length in each of the four suits. There are four possibilities to first identify the three-card suit and three possibilities to next identify the doubleton. Hence, the number of suit permutations of the 4-4-3-2 pattern is twelve. Or, stated differently, in total there are twelve ways a 4-4-3-2 pattern can be mapped onto the four suits.

Below table lists all 39 possible hand patterns, their probability of occurrence, as well as the number of suit permutations for each pattern. The list is ordered according to likelihood of occurrence of the hand patterns.[3]

Pattern Probability #
4-4-3-2 0.2155 12
5-3-3-2 0.1552 12
5-4-3-1 0.1293 24
5-4-2-2 0.1058 12
4-3-3-3 0.1054 4
6-3-2-2 0.0564 12
6-4-2-1 0.0470 24
6-3-3-1 0.0345 12
5-5-2-1 0.0317 12
4-4-4-1 0.0299 4
7-3-2-1 0.0188 24
6-4-3-0 0.0133 24
5-4-4-0 0.0124 12
Pattern Probability #
5-5-3-0 0.0090 12
6-5-1-1 0.0071 12
6-5-2-0 0.0065 24
7-2-2-2 0.0051 4
7-4-1-1 0.0039 12
7-4-2-0 0.0036 24
7-3-3-0 0.0027 12
8-2-2-1 0.0019 12
8-3-1-1 0.0012 12
7-5-1-0 0.0011 24
8-3-2-0 0.0011 24
6-6-1-0 0.00072 12
8-4-1-0 0.00045 24
Pattern Probability #
9-2-1-1 0.00018 12
9-3-1-0 0.00010 24
9-2-2-0 0.000082 12
7-6-0-0 0.000056 12
8-5-0-0 0.000031 12
10-2-1-0 0.000011 24
9-4-0-0 0.000010 12
10-1-1-1 0.000004 4
10-3-0-0 0.0000015 12
11-1-1-0 0.0000002 12
11-2-0-0 0.0000001 12
12-1-0-0 0.000000003 12
13-0-0-0 0.000000000006 4

The 39 hand patterns can by classified into four hand types: balanced hands, three-suiters, two suiters and single suiters. Below table gives the a priori likelihoods of being dealt a certain hand-type.

Hand type Patterns Probability
Balanced 4-3-3-3, 4-4-3-2, 5-3-3-2 0.4761
Two-suiter 5-4-2-2, 5-4-3-1, 5-5-2-1, 5-5-3-0, 6-5-1-1, 6-5-2-0, 6-6-1-0, 7-6-0-0 0.2902
Single-suiter 6-3-2-2, 6-3-3-1, 6-4-2-1, 6-4-3-0, 7-2-2-2, 7-3-2-1, 7-3-3-0, 7-4-1-1, 7-4-2-0, 7-5-1-0, 8-2-2-1, 8-3-1-1, 8-3-2-0, 8-4-1-0, 8-5-0-0, 9-2-1-1, 9-2-2-0, 9-3-1-0, 9-4-0-0, 10-1-1-1, 10-2-1-0, 10-3-0-0, 11-1-1-0, 11-2-0-0, 12-1-0-0, 13-0-0-0 0.1915
Three-suiter 4-4-4-1, 5-4-4-0 0.0423

Alternative grouping of the 39 hand patterns can be made either by longest suit or by shortest suit. Below tables gives the a priori chance of being dealt a hand with a longest or a shortest suit of given length.

Longest suit Patterns Probability
4 card 4-3-3-3, 4-4-3-2, 4-4-4-1 0.3508
5 card 5-3-3-2, 5-4-2-2, 5-4-3-1, 5-5-2-1, 5-4-4-0, 5-5-3-0 0.4434
6 card 6-3-2-2, 6-3-3-1, 6-4-2-1, 6-4-3-0, 6-5-1-1, 6-5-2-0, 6-6-1-0 0.1655
7 card 7-2-2-2, 7-3-2-1, 7-3-3-0, 7-4-1-1, 7-4-2-0, 7-5-1-0, 7-6-0-0 0.0353
8 card 8-2-2-1, 8-3-1-1, 8-3-2-0, 8-4-1-0, 8-5-0-0 0.0047
9 card 9-2-1-1, 9-2-2-0, 9-3-1-0, 9-4-0-0 0.00037
10 card 10-1-1-1, 10-2-1-0, 10-3-0-0 0.000017
11 card 11-1-1-0, 11-2-0-0 0.0000003
12 card 12-1-0-0 0.000000003
13 card 13-0-0-0 0.000000000006
Shortest suit Patterns Probability
Three card 4-3-3-3 0.1054
Doubleton 4-4-3-2, 5-3-3-2, 5-4-2-2, 6-3-2-2, 7-2-2-2 0.5380
Singleton 4-4-4-1, 5-4-3-1, 5-5-2-1, 6-3-3-1, 6-4-2-1, 6-5-1-1, 7-3-2-1, 7-4-1-1, 8-2-2-1, 8-3-1-1, 9-2-1-1, 10-1-1-1 0.3055
Void 5-4-4-0, 5-5-3-0, 6-4-3-0, 6-5-2-0, 6-6-1-0, 7-3-3-0, 7-4-2-0, 7-5-1-0, 7-6-0-0, 8-3-2-0, 8-4-1-0, 8-5-0-0, 9-2-2-0, 9-3-1-0, 9-4-0-0, 10-2-1-0, 10-3-0-0, 11-1-1-0, 11-2-0-0, 12-1-0-0, 13-0-0-0 0.0512

## Number of possible deals

In total there are 53,644,737,765,488,792,839,237,440,000 (5.36 x 1028) different deals possible, which is equal to $52!/(13!)^4$. The immenseness of this number can be understood by answering the question "How large an area would you need to spread all possible bridge deals if each deal would occupy only one square millimeter?". The answer is: an area more than a hundred million times the total area of Earth.

Obviously, the deals that are identical except for swapping—say—the 2 and the 3 would be unlikely to give a different result. To make the irrelevance of small cards explicit (which is not always the case though), in bridge such small cards are generally denoted by an 'x'. Thus, the "number of possible deals" in this sense depends of how many non-honour cards (2, 3, .. 9) are considered 'indistinguishable'. For example, if 'x' notation is applied to all cards smaller than ten, then the suit distributions A987-K106-Q54-J32 and A432-K105-Q76-J98 would be considered identical.

The table below [4] gives the number of deals when various numbers of small cards are considered indistinguishable.

Suit composition Number of deals
AKQJT9876543x 53,644,737,765,488,792,839,237,440,000
AKQJT987654xx 7,811,544,503,918,790,990,995,915,520
AKQJT98765xxx 445,905,120,201,773,774,566,940,160
AKQJT9876xxxx 14,369,217,850,047,151,709,620,800
AKQJT987xxxxx 314,174,475,847,313,213,527,680
AKQJT98xxxxxx 5,197,480,921,767,366,548,160
AKQJT9xxxxxxx 69,848,690,581,204,198,656
AKQJTxxxxxxxx 800,827,437,699,287,808
AKQJxxxxxxxxx 8,110,864,720,503,360
AKQxxxxxxxxxx 74,424,657,938,928
AKxxxxxxxxxxx 630,343,600,320
Axxxxxxxxxxxx 4,997,094,488
xxxxxxxxxxxxx 37,478,624

Note that the last entry in the table (37,478,624) corresponds to the number of different distributions of the deck (the number of deals when cards are only distinguished by their suit).

## Probability of Losing-Trick Counts

The Losing-Trick Count is an alternative to the HCP count as a method of hand evaluation.

LTC Number of Hands Probability
0 4,245,032 0.000668%
1 90,206,044 0.0142%
2 872,361,936 0.137%
3 5,080,948,428 0.8%
4 19,749,204,780 3.11%
5 53,704,810,560 8.46%
6 104,416,332,340 16.4%
7 145,971,648,360 23.0%
8 145,394,132,760 22.9%
9 100,454,895,360 15.8%
10 45,618,822,000 7.18%
11 12,204,432,000 1.92%
12 1,451,520,000 0.229%
13 0 0%

## References

1. ^ a b "Mathematical Tables" (Table 4). Francis, Henry G., Editor-in-Chief; Truscott, Alan F., Executive Editor; Francis, Dorthy A., Editor, Fifth Edition (1994). The Official Encyclopedia of Bridge (5th ed.). Memphis, TN: American Contract Bridge League. p. 278. ISBN 0-943855-48-9. LCCN 96188639.
2. ^ Richard Pavlicek. "High Card Expectancy." link
3. ^ Richard Pavlicek. "Against All Odds." link
4. ^ Counting Bridge Deals, Jeroen Warmerdam