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.
This table represents the different ways that two to thirteen particular cards may be distributed, or may lie or split, between two unknown 13card 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  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~ 
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. To find the likelihood of a certain point range, one simply subtracts the two relevant cumulative probabilities. So, the likelihood of being dealt a 1219 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 
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 4432 pattern consisting of two fourcard suits, a threecard suit and a doubleton.
Note that the hand pattern leaves unspecified which particular suits contain the indicated lengths. For a 4432 pattern, one needs to specify which suit contains the threecard 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 threecard suit and three possibilities to next identify the doubleton. Hence, the number of suit permutations of the 4432 pattern is twelve. Or, stated differently, in total there are twelve ways a 4432 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.



The 39 hand patterns can by classified into four hand types: balanced hands, threesuiters, two suiters and single suiters. Below table gives the a priori likelihoods of being dealt a certain handtype.
Hand type  Patterns  Probability 

Balanced  4333, 4432, 5332  0.4761 
Twosuiter  5422, 5431, 5521, 5530, 6511, 6520, 6610, 7600  0.2902 
Singlesuiter  6322, 6331, 6421, 6430, 7222, 7321, 7330, 7411, 7420, 7510, 8221, 8311, 8320, 8410, 8500, 9211, 9220, 9310, 9400, 10111, 10210, 10300, 11110, 11200, 12100, 13000  0.1915 
Threesuiter  4441, 5440  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  4333, 4432, 4441  0.3508 
5 card  5332, 5422, 5431, 5521, 5440, 5530  0.4434 
6 card  6322, 6331, 6421, 6430, 6511, 6520, 6610  0.1655 
7 card  7222, 7321, 7330, 7411, 7420, 7510, 7600  0.0353 
8 card  8221, 8311, 8320, 8410, 8500  0.0047 
9 card  9211, 9220, 9310, 9400  0.00037 
10 card  10111, 10210, 10300  0.000017 
11 card  11110, 11200  0.0000003 
12 card  12100  0.000000003 
13 card  13000  0.000000000006 
Shortest suit  Patterns  Probability 

Three card  4333  0.1054 
Doubleton  4432, 5332, 5422, 6322, 7222  0.5380 
Singleton  4441, 5431, 5521, 6331, 6421, 6511, 7321, 7411, 8221, 8311, 9211, 10111  0.3055 
Void  5440, 5530, 6430, 6520, 6610, 7330, 7420, 7510, 7600, 8320, 8410, 8500, 9220, 9310, 9400, 10210, 10300, 11110, 11200, 12100, 13000  0.0512 
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 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 nonhonour cards (2, 3, .. 9) are considered 'indistinguishable'. For example, if 'x' notation is applied to all cards smaller than ten, then the suit distributions A987K106Q54J32 and A432K105Q76J98 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 LosingTrick 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% 