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The statistics that are derived from all the 39 types of hand distributions in Bridge are very important to get the best result in the long run. A must know for any good bridge player. Not only useful when you design a bidding system but also in declarer play, defence and guessing what your partner might have.

There are more than 600 trillion possibilities of getting a set of 13 cards when you play bridge. So, unless we already play more than 600 trillion times, the chance is that we will never see most of them.

But the good thing is, we don’t really have to deal with 650 trillion kinds of rules/guideline as they can be categorized into only 39 types of hands distribution: from the super balance 4-3-3-3 to 13-0-0-0.

Sorted by Distribution: The 39 Types of Hand Distributions

Distribution Total HandsProbability
1:13 – 0 – 0 – 04<0.1%
2:12 – 1 – 0 – 02,028<0.1%
3:11 – 2 – 0 – 073,008<0.1%
4:11 – 1 – 1 – 0158,184<0.1%
5:10 – 3 – 0 – 0981,552<0.1%
6:10 – 2 – 1 – 06,960,096<0.1%
7:10 – 1 – 1 – 12,513,368<0.1%
8:9 – 4 – 0 – 06,134,700<0.1%
9:9 – 3 – 1 – 063,800,880<0.1%
10:9 – 2 – 2 – 052,200,720<0.1%
11:9 – 2 – 1 – 1113,101,560<0.1%
12:8 – 5 – 0 – 019,876,428<0.1%
13:8 – 4 – 1 – 0287,103,960<0.1%
14:8 – 3 – 2 – 0689,049,5040.1%
15:8 – 3 – 1 – 1746,470,2960.1%
16:8 – 2 – 2 – 11,221,496,8480.2%
17:7 – 6 – 0 – 035,335,872<0.1%
18:7 – 5 – 1 – 0689,049,5040.1%
19:7 – 4 – 2 – 02,296,831,6800.4%
20:7 – 4 – 1 – 12,488,234,3200.4%
21:7 – 3 – 3 – 01,684,343,2320.3%
22:7 – 3 – 2 – 111,943,524,7361.9%
23:7 – 2 – 2 – 23,257,324,9280.5%
24:6 – 6 – 1 – 0459,366,3360.1%
25:6 – 5 – 2 – 04,134,297,0240.7%
26:6 – 5 – 1 – 14,478,821,7760.7%
27:6 – 4 – 3 – 08,421,716,1601.3%
28:6 – 4 – 2 – 129,858,811,8404.7%
29:6 – 3 – 3 – 121,896,462,0163.4%
30:6 – 3 – 2 – 235,830,574,2085.6%
31:5 – 5 – 3 – 05,684,658,4080.9%
26:8 – 4 – 1 – 0287,103,960<0.1%
27:9 – 2 – 1 – 1113,101,560<0.1%
28:9 – 3 – 1 – 063,800,880<0.1%
29:9 – 2 – 2 – 052,200,720<0.1%
30:7 – 6 – 0 – 035,335,872<0.1%
31:8 – 5 – 0 – 019,876,428<0.1%
32:5 – 5 – 2 – 120,154,697,9923.2%
33:5 – 4 – 4 – 07,895,358,9001.2%
34:5 – 4 – 3 – 182,111,732,56012.9%
35:5 – 4 – 2 – 267,182,326,64010.6%
36:5 – 3 – 3 – 298,534,079,07215.5%
37:4 – 4 – 4 – 119,007,345,5003.0%
38:4 – 4 – 3 – 2136,852,887,60021.6%
39:4 – 3 – 3 – 366,905,856,16010.5%
 Total635,013,559,600 

Sorted by Percentage: The 39 Types of Hand Distributions

Distribution Total HandsProbability
1:4 – 4 – 3 – 2136,852,887,60021.6%
2:5 – 3 – 3 – 298,534,079,07215.5%
3:5 – 4 – 3 – 182,111,732,56012.9%
4:5 – 4 – 2 – 267,182,326,64010.6%
5:4 – 3 – 3 – 366,905,856,16010.5%
6:6 – 3 – 2 – 235,830,574,2085.6%
7;6 – 4 – 2 – 129,858,811,8404.7%
8;6 – 3 – 3 – 121,896,462,0163.4%
9:5 – 5 – 2 – 120,154,697,9923.2%
10:4 – 4 – 4 – 119,007,345,5003.0%
11:7 – 3 – 2 – 111,943,524,7361.9%
12:6 – 4 – 3 – 08,421,716,1601.3%
13:5 – 4 – 4 – 07,895,358,9001.2%
14:5 – 5 – 3 – 05,684,658,4080.9%
15:6 – 5 – 1 – 14,478,821,7760.7%
16:6 – 5 – 2 – 04,134,297,0240.7%
17:7 – 2 – 2 – 23,257,324,9280.5%
18:7 – 4 – 1 – 12,488,234,3200.4%
19:7 – 4 – 2 – 02,296,831,6800.4%
20:7 – 3 – 3 – 01,684,343,2320.3%
21:8 – 2 – 2 – 11,221,496,8480.2%
22:8 – 3 – 1 – 1746,470,2960.1%
23:8 – 3 – 2 – 0689,049,5040.1%
24:7 – 5 – 1 – 0689,049,5040.1%
25:6 – 6 – 1 – 0459,366,3360.1%
32:10 – 2 – 1 – 06,960,096<0.1%
33:9 – 4 – 0 – 06,134,700<0.1%
34:10 – 1 – 1 – 12,513,368<0.1%
35;10 – 3 – 0 – 0981,552<0.1%
36:11 – 1 – 1 – 0158,184<0.1%
37:11 – 2 – 0 – 073,008<0.1%
38:12 – 1 – 0 – 02,028<0.1%
39:13 – 0 – 0 – 04<0.1%
 Total635,013,559,600 

References: Bill Butler, “Durango Bill’s Bridge Probabilities and Combinatorics”, Richard Pavlicek, “Against All Odds”

Some of the interesting finding we can observe from the tables above:

  • The most common distribution is 4-4-3-2 with 21.6% odds.
  • Despite the flattest, 4-3-3-3 distribution in only the fifth most common distribution with 10.5% odds.
  • Long (and extremely long) suit is hardest to get

The 39 Types of Hand Distributions Into 5 Groups

Don’t feel overwhelmed! To make it even easier, we can manage the 39 types of hand distribution into 5 groups categories. In no particular order, they are:

1. Balanced hand (Total 47.6%)

 

Distribution Total HandsProbability
1:5 – 3 – 3 – 298,534,079,07215.5%
2:4 – 4 – 3 – 2136,852,887,60021.6%
3:4 – 3 – 3 – 366,905,856,16010.5%
TOTAL:47.6%

You probably argue that 5-3-3-2 is not balanced. Yes, it’s more accurate to be defined as “semi-balanced”. But what you rebid with that hand? We need to treat 5-3-3-2 as balanced as you cannot really re-bid with another suit.

A rebid of the same suit should promise at least an additional card of that suit. But there is no other 4-cards suit to bid. Hence, treating it as balanced similar to 4-3-3-3 is probably the best treatment.

This category is the biggest group of all. It accounts of almost half of all the hand that you play. So, very familiar with the bidding for balanced hand should be the priority of your bidding system. If you do that first , at least 1 out of 2 hands that you play will be covered.

Please note that the highest hand probability of distribution is not 4-3-3-3 (only 10.5%) but 4-4-3-2 (21.6%). So, start your bidding system discussion with your partner on balanced hand sequence first!

Note that category (3.3.1) below (5422 distribution) may have similarity in regard to bidding the distribution.

2. Three Suiter Hand (Total 4.24%)

 

Distribution Total HandsProbability
1:5 – 4 – 4 – 07,895,358,9001.2%
2:4 – 4 – 4 – 119,007,345,5003.0%
TOTAL:4.24%

Three suiters hands are quite notoriously difficult to bid/rebid during “normal” bidding and sometimes the partnership may miss the fit on the third suits. Therefore, some bidding system is actually treating this distribution seriously and creating a special opening bid or sequence just to cater to this type of distribution.

The odds of happening is one for every 25 boards.

Note that category (3.3.2) below (5431 distribution) may have similarity in regard to bidding the distribution.

3. Two-Suiter Hand

3.1. Two-suiter hand with 54+ (Total: 35.90%)

 

Distribution Total HandsProbability
1:9 – 4 – 0 – 06,134,700<0.1%
2:8 – 5 – 0 – 019,876,428<0.1%
3:8 – 4 – 1 – 0287,103,960<0.1%
4:7 – 6 – 0 – 035,335,872<0.1%
5:7 – 5 – 1 – 0689,049,5040.1%
6:7 – 4 – 2 – 02,296,831,6800.4%
7:7 – 4 – 1 – 12,488,234,3200.4%
8:6 – 6 – 1 – 0459,366,3360.1%
9:6 – 5 – 2 – 04,134,297,0240.7%
10:6 – 5 – 1 – 14,478,821,7760.7%
11:6 – 4 – 3 – 08,421,716,1601.3%
12:6 – 4 – 2 – 129,858,811,8404.7%
13:5 – 5 – 3 – 05,684,658,4080.9%
14:5 – 5 – 2 – 120,154,697,9923.2%
15:5 – 4 – 3 – 182,111,732,56012.9%
16:5 – 4 – 2 – 267,182,326,64010.6%
TOTAL:35.90%

Any hand distribution with 5+cards in 1 suit and 4+ cards in another suit, I consider them as two-suiter hand.

This group is responsible for more than 1/3 of all hand that you play. Hence, careful bidding as to how to rebid both suits is very important to make sure you play at the correct level. 

3.2. Two-Suiter hand with 55+ (Total: 5.7+%)

 

Distribution Total HandsProbability
1:8 – 5 – 0 – 019,876,428<0.1%
2:7 – 6 – 0 – 035,335,872<0.1%
3:7 – 5 – 1 – 0689,049,5040.1%
4:6 – 6 – 1 – 0459,366,3360.1%
5:6 – 5 – 2 – 04,134,297,0240.7%
6:6 – 5 – 1 – 14,478,821,7760.7%
7:5 – 5 – 3 – 05,684,658,4080.9%
8:5 – 5 – 2 – 120,154,697,9923.2%
TOTAL:>5.7%

If we narrow down the 2-suiter hand to have a minimum of 5-5 suit, the table would be as on the right.

3.3. Two-Suiter hand with 5+4

The distribution with 4-cards second suit and longer first suit can be re-group as follows:

  1. (5422) can be grouped as (semi) balanced (Total: 10.6%)
  2. (5431) can be grouped as (almost) 3-suiter (Total: 12.9%)
  3. (6+4xx) can be grouped as 1-suiter (Total: 6.9%)

4. One-Suiter Hand (Total: 12.17%)

 

Distribution Total HandsProbability
1:8 – 3 – 2 – 0689,049,5040.1%
2:8 – 3 – 1 – 1746,470,2960.1%
3:8 – 2 – 2 – 11,221,496,8480.2%
4:7 – 3 – 3 – 01,684,343,2320.3%
5:7 – 3 – 2 – 111,943,524,7361.9%
6:7 – 2 – 2 – 23,257,324,9280.5%
7:6 – 3 – 3 – 121,896,462,0163.4%
8:6 – 3 – 2 – 235,830,574,2085.6%
TOTAL:12.17%

One-suiter hand with 6 to 8 cards in one hand account for 12.17% of the possibility. That’s roughly 1 for every 8 boards.

Note that category (3.3.3) above (6+4xx distribution) may have similarity in regard to bidding the distribution.

5. Extreme 1 Suiter (Total: 0.04%)

 

Distribution Total HandsProbability
1:13 – 0 – 0 – 04<0.1%
2:12 – 1 – 0 – 02,028<0.1%
3:11 – 2 – 0 – 073,008<0.1%
4:11 – 1 – 1 – 0158,184<0.1%
5:10 – 3 – 0 – 0981,552<0.1%
6:10 – 2 – 1 – 06,960,096<0.1%
7:10 – 1 – 1 – 12,513,368<0.1%
8:9 – 3 – 1 – 063,800,880<0.1%
9:9 – 2 – 2 – 052,200,720<0.1%
10:9 – 2 – 1 – 1113,101,560<0.1%
TOTAL:0.04%

There are 10 types of distribution that include a nine-carder hand. The statistical probability is similar to play 2500 boards before you meet one.

There is more to the 39 types of hand distributions

Further detail of what presented here can be further explored and study. You might not see the importance now. But in one of those moments in the future, you will just know:

39 types of hand distributions