Bridge probabilities
Encyclopedia
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 a priori
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.
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
.
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.
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.
The 39 hand patterns can by classified into four hand types: balanced hand
s, three-suiters, two suiter
s and single suiter
s. Below table gives the a priori likelihoods of being dealt a certain hand-type.
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.
. 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
.
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.
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).
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 prioriA 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 prioriA 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.
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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 earthOrders 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).