Contract 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	2 - 0	0.48	2	0.24
3	2 - 1	0.78	6	0.13
3	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 surface 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 2 "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.
↑ Richard Pavlicek. "High Card Expectancy." link
↑ Richard Pavlicek. "Against All Odds." link
↑ Counting Bridge Deals, Jeroen Warmerdam

Émile, Borel; André, Chéron (1940). Théorie Mathématique du Bridge. Gauthier-Villars. Second French edition by the authors in 1954. Translated and edited into English by Alec Traub as The Mathematical Theory of Bridge; printed in 1974 in Taiwan through the assistance of C.C. Wei.
Kelsey, Hugh; Glauert, Michael (1980). Bridge Odds for Practical Players. Master Bridge Series. London: Victor Gollancz Ltd in association with Peter Crawley. ISBN 0-575-02799-1.
Reese, Terence; Trézel, Roger (1986). Master the Odds in Bridge. Master Bridge Series. London: Victor Gollancz Ltd in association with Peter Crawley. ISBN 0-575-02597-2.

Contract bridge articles

Overview

General

Bidding

General	Balanced hand Balancing (bridge) Bidding box Bidding system Board (bridge) Bridge convention Brown sticker Convention card Cue bid Five-card majors Forcing bid Forcing pass Game try Hand evaluation Honor point count Inverted minors Law of total tricks Losing-Trick Count Major suit Minor suit Optimum contract and par contract Overcall Preempt Prepared opening bid Principle of fast arrival Psychic bid Quantitative no trump bids Reverse (bridge) Sacrifice (bridge) Shooting (bridge) Single suiter Strong pass Takeout double Three suiter Two suiter Useful space principle Void (cards) Weak two bid Zar Points

Systems	List of bidding systems 2/1 game forcing Acol Bidding system Blue Club Boring Club Bridge Base Basic Bridge World Standard Canapé (bridge) Carrot Club EHAA Fantunes Highly unusual method Kaplan–Sheinwold Little Major Moscito OKbridge 2/1 Polish Club Precision Club Roman Club Romex system Säffle Spade Standard American Strong club system

Conventions	List of bidding conventions

Card play

General	List of play techniques Avoidance play Beer card Caddy Card reading Duck Endplay Entry Grosvenor gambit Hold up Percentage play Pin (bridge) Principle of restricted choice Probabilities Revoke Ruff Shooting Smother play Tempo Trump promotion Uppercut Vacant Places

Declarer play	Basic: Finesse Safety play Coups: Alcatraz coup Bath coup Belladonna coup Coup Coup en passant Crocodile coup Deschapelles coup Devil's coup Loser on loser Merrimac coup Morton's fork coup Scissors coup Trump coup Vienna coup Squeezes: Backwash squeeze Cannibal squeeze Clash squeeze Compound squeeze Criss-cross squeeze Double squeeze Entry squeeze Entry-shifting squeeze Guard squeeze Knockout squeeze Non-simultaneous double squeeze Progressive squeeze Pseudo-squeeze Saturated squeeze Simple squeeze Simultaneous double squeeze Single-suit squeeze Squeeze play Stepping-stone squeeze Strip squeeze Triple squeeze Trump squeeze Vice squeeze Winkle squeeze Suit combinations: Suit combination Suit combination – J missing Suit combination – 10 missing Suit combinations – K10 missing

Defender play	Forcing defense Journalist leads Opening lead Rule of 10-12 Rule of 11 Rusinow leads Signal Smith signal

People and organizations

General	ACBL Youngest Life Master List of contract bridge people List of nationality transfers in sport

Players by country	Australia Austria Brazil Britain Bulgaria Canada China Denmark Netherlands Egypt Fiction France Germany Hungary India Ireland Israel Italy Jewish players Mexico Monaco New Zealand Norway Pakistan Poland Russia Sweden Swirzerlad Taiwan United States

Other lists	Alphabetical list of contract bridge people with Wikipedia (english) articles List of bridge administrators List of bridge writers

Teams	Blue Team Dallas Aces Four Aces

Clubs	Bridge Club Brașov Cavendish Club Crockford's Galatasaray Bridge Team Melville Bridge Club Portland Club (London) Savoy Club Young Chelsea Bridge Club

Governing bodies	American Bridge Association American Contract Bridge League Austrian Bridge Federation Brazilian Bridge Federation British Bridge League Canadian Bridge Federation Dutch Bridge Federation English Bridge Union European Bridge League Hungarian Bridge Federation International Mind Sports Association List of bridge governing bodies Norwegian Bridge Federation World Bridge Federation United States Bridge Federation

Championships

General	List of bridge competitions and awards

World	Bermuda Bowl Bridge at the 2008 World Mind Sports Games Bridge at the 2012 World Mind Sports Games Cavendish Invitational Computer Olympiad McConnell Cup Rosenblum Cup Senior Bowl (bridge) Triple crown of bridge Venice Cup World Bridge Championships World IMP Pairs Championship World Junior Pairs Championship World Junior Teams Championship World Mind Sports Games World Mixed Pairs Championship World Mixed Teams Championship World Open Pairs Championship World Senior Pairs Championship World Senior Teams Championship World Team Olympiad World Transnational Open Teams Championship World Women Pairs Championship

National and Zonal	Buffett Cup Camrose Trophy Commonwealth Nations Bridge Championships European Universities Bridge Championships Gold Cup (bridge) North American bridge Championships: ACBL King or Queen of Bridge Blue Ribbon Pairs Bruce LM-5000 Pairs Chicago Mixed Board-a-Match Fall National Open Pairs Fast Open Pairs Fishbein Trophy Goren Trophy Grand National Teams Hilliard Mixed Pairs Jacoby Open Swiss Teams Keohane North American Swiss Teams Lebhar IMP Pairs Leventritt Silver Ribbon Pairs Machlin Women's Swiss Teams Manfield Non-Life Master Pairs Marcus Cup Master Individual Mini-Blue Ribbon Pairs Mini-Spingold Mitchell Board-a-Match Teams Mott-Smith Trophy Nail Life Master Open Pairs National 199er Pairs National 49er Pairs National 99er Pairs Non-Life Master Swiss Teams Norman Kay Platinum Pairs North American Bridge Championships North American Pairs Red Ribbon Pairs Reisinger Rockwell Mixed Pairs Roth Open Swiss Teams Senior Knockout Teams Silodor Open Pairs Smith Life Master Women's Pairs Spingold Sternberg Women's Board-a-Match Teams Truscott Senior Swiss Teams Vanderbilt Trophy Von Zedtwitz Life Master Pairs Wagar Women's Knockout Teams Wernher Open Pairs Whitehead Women's Pairs Young LM-1500 Pairs

Publications and resources

Books	List of contract bridge books 25 Bridge Conventions You Should Know Bridge Squeezes Complete The Cardturner Design for Bidding Contract Bridge for Beginners Edwin Kantar bibliography Master Point Press Planning the Play of a Bridge Hand Terence Reese bibliography The Official Encyclopedia of Bridge Tickets to the Devil

Magazines	List of bridge magazines Bridge d'Italia Bridge Magazine The Bridge World

External links	Bridge Base Basic Bridge Base Inc. OK bridge 2/1

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