Bridge probabilities
Encyclopedia
In the game of bridge
Contract bridge
Contract bridge, usually known simply as bridge, is a trick-taking card game using a standard deck of 52 playing cards played by four players in two competing partnerships with partners sitting opposite each other around a small table...

 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 a priori
A priori (statistics)
In statistics, a priori knowledge is prior knowledge about a population, rather than that estimated by recent observation. It is common in Bayesian inference to make inferences conditional upon this knowledge, and the integration of a priori knowledge is the central difference between the Bayesian...

probabilities, i.e. the probabilities in the absence of any further information. During bidding and play, more information of hands becomes available and requires players to change their probability assumptions.

Probability of suit distributions in two hidden hands

This table 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
Vacant Places
In the card game bridge, the law or principle of vacant places is a simple method for estimating the probable location of any particular card in the four hands...

.
#cards Distribution Probability Combinations Individual Prob
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
A priori (statistics)
In statistics, a priori knowledge is prior knowledge about a population, rather than that estimated by recent observation. It is common in Bayesian inference to make inferences conditional upon this knowledge, and the integration of a priori knowledge is the central difference between the Bayesian...

 probabilities that a given hand contains no more than a specified number of hcp is given in the table below. 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.
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
A priori (statistics)
In statistics, a priori knowledge is prior knowledge about a population, rather than that estimated by recent observation. It is common in Bayesian inference to make inferences conditional upon this knowledge, and the integration of a priori knowledge is the central difference between the Bayesian...

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 permuatation for each pattern. The list is ordered according to likelihood of occurrence of the hand patterns.
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 hand
Balanced hand
In the game of bridge a balanced hand denotes a hand containing no singleton or void and at most one doubleton. As a bridgehand contains thirteen cards, only three hand patterns can be classified as balanced: 4-3-3-3, 4-4-3-2 and 5-3-3-2...

s, three-suiters, two suiter
Two suiter
In contract bridge, a two suiter is a hand containing cards mostly from two of the four suits. Traditionally a hand is considered a two suiter if it contains at least ten cards in two suits, with the two suits not differing in length by more than one card. Depending on suit quality and partnership...

s and single suiter
Single suiter
In contract bridge, a single suiter is a hand containing at least six cards in one suit and with all other suits being at least two cards shorter than this longest suit. Many hand patterns can be classified as single suiters. Typical examples are 6-3-2-2, 6-3-3-1 and 7-3-2-1...

s. 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 10^28) different deals possible, which is equal to . 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 the earth
Orders of magnitude (area)
This page is a progressive and labeled list of the SI area orders of magnitude, with certain examples appended to some list objects.-References:...

.

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 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).
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
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