Bridge probabilities
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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.
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[edit] Probability of suit distributions in two hidden hands
The a priori probabilities that a specific number of cards outstanding in a suit split in the various ways over two hidden hands is given in the table below[1].
| #cards | Distribution | Probability |
|---|---|---|
| 2 | 1 - 1 | 0.52 |
| 2 - 0 | 0.48 | |
| 3 | 2 - 1 | 0.78 |
| 3 - 0 | 0.22 | |
| 4 | 2 - 2 | 0.41 |
| 3 - 1 | 0.50 | |
| 4 - 0 | 0.10 | |
| 5 | 3 - 2 | 0.68 |
| 4 - 1 | 0.28 | |
| 5 - 0 | 0.04 | |
| 6 | 3 - 3 | 0.36 |
| 4 - 2 | 0.48 | |
| 5 - 1 | 0.15 | |
| 6 - 0 | 0.01 | |
| 7 | 4 - 3 | 0.62 |
| 5 - 2 | 0.31 | |
| 6 - 1 | 0.07 | |
| 7 - 0 | 0.01 | |
| 8 | 4 - 4 | 0.33 |
| 5 - 3 | 0.47 | |
| 6 - 2 | 0.17 | |
| 7 - 1 | 0.03 | |
| 8 - 0 | 0.00 |
[edit] Probability of HCP distribution
The a priori probabilities that a given hand contains no more than a specified number of high card points (hcp) is given in below table[1]. To find the likelihood of a certain hcp-range one simply subtracts the two relevant cumulative probabilities. So, the likelihood of getting 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 | ||
|---|---|---|---|---|---|---|---|
| 0 | 0.004 | 8 | 0.375 | 16 | 0.936 | ||
| 1 | 0.012 | 9 | 0.469 | 17 | 0.959 | ||
| 2 | 0.025 | 10 | 0.563 | 18 | 0.975 | ||
| 3 | 0.050 | 11 | 0.652 | 19 | 0.986 | ||
| 4 | 0.088 | 12 | 0.732 | 20 | 0.992 | ||
| 5 | 0.140 | 13 | 0.802 | 21 | 0.996 | ||
| 6 | 0.206 | 14 | 0.858 | 22 | 0.998 | ||
| 7 | 0.286 | 15 | 0.903 | 23 | 0.999 |
[edit] 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. Most of these are unlikely to occur. Only five patterns have an a priori probability exceeding 10%. 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 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 |
[edit] Number of possible deals
In total there are 53,644,737,765,488,792,839,237,440,000 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 the earth.
Obviously, the deals that are identical except for swapping—say—the ♥2 and the ♥3 would unlikely 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 [2] 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 distinghuished by their suit).
[edit] References
- ^ a b H.G. Francis, A.F. Truscott and D.A. Francis (Eds.): The Official Encyclopedia of Bridge, 5th Edition, ISBN 0-943855-48-9.
- ^ Counting Bridge Deals, Jeroen Warmerdam
