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.
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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 |
Read more about this topic: Bridge Probabilities
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