Passion of The Game of Bridge

39 Type of Hand Distribution in Bridge

39 Type of Hand Distribution in Bridge

There are more than 600 trillion possibility of getting a set of 13 cards when you play bridge. SO, the chance is that we will never see all of them unless we play much bigger number than 600 trillion games. But the good thing is, we don’t really have to deal with 650 trillion kind of bidding as they can be categorize into only 39 type of hands distribution: from the super balance 4-3-3-3 to 13-0-0-0.

All Possible Hand in Bridge

Distribution Sorted by Suit

 

Distribution Sorted by Probability

 

Distribution Total Hands Probability Distribution Total Hands Probability
1: 13 – 0 – 0 – 0 4 <0.1% 1: 4 – 4 – 3 – 2 136,852,887,600 21.6%
2: 12 – 1 – 0 – 0 2,028 <0.1% 2: 5 – 3 – 3 – 2 98,534,079,072 15.5%
3: 11 – 2 – 0 – 0 73,008 <0.1% 3: 5 – 4 – 3 – 1 82,111,732,560 12.9%
4: 11 – 1 – 1 – 0 158,184 <0.1% 4: 5 – 4 – 2 – 2 67,182,326,640 10.6%
5: 10 – 3 – 0 – 0 981,552 <0.1% 5: 4 – 3 – 3 – 3 66,905,856,160 10.5%
6: 10 – 2 – 1 – 0 6,960,096 <0.1% 6: 6 – 3 – 2 – 2 35,830,574,208 5.6%
7: 10 – 1 – 1 – 1 2,513,368 <0.1% 7; 6 – 4 – 2 – 1 29,858,811,840 4.7%
8: 9 – 4 – 0 – 0 6,134,700 <0.1% 8; 6 – 3 – 3 – 1 21,896,462,016 3.4%
9: 9 – 3 – 1 – 0 63,800,880 <0.1% 9: 5 – 5 – 2 – 1 20,154,697,992 3.2%
10: 9 – 2 – 2 – 0 52,200,720 <0.1% 10: 4 – 4 – 4 – 1 19,007,345,500 3.0%
11: 9 – 2 – 1 – 1 113,101,560 <0.1% 11: 7 – 3 – 2 – 1 11,943,524,736 1.9%
12: 8 – 5 – 0 – 0 19,876,428 <0.1% 12: 6 – 4 – 3 – 0 8,421,716,160 1.3%
13: 8 – 4 – 1 – 0 287,103,960 <0.1% 13: 5 – 4 – 4 – 0 7,895,358,900 1.2%
14: 8 – 3 – 2 – 0 689,049,504 0.1% 14: 5 – 5 – 3 – 0 5,684,658,408 0.9%
15: 8 – 3 – 1 – 1 746,470,296 0.1% 15: 6 – 5 – 1 – 1 4,478,821,776 0.7%
16: 8 – 2 – 2 – 1 1,221,496,848 0.2% 16: 6 – 5 – 2 – 0 4,134,297,024 0.7%
17: 7 – 6 – 0 – 0 35,335,872 <0.1% 17: 7 – 2 – 2 – 2 3,257,324,928 0.5%
18: 7 – 5 – 1 – 0 689,049,504 0.1% 18: 7 – 4 – 1 – 1 2,488,234,320 0.4%
19: 7 – 4 – 2 – 0 2,296,831,680 0.4% 19: 7 – 4 – 2 – 0 2,296,831,680 0.4%
20: 7 – 4 – 1 – 1 2,488,234,320 0.4% 20: 7 – 3 – 3 – 0 1,684,343,232 0.3%
21: 7 – 3 – 3 – 0 1,684,343,232 0.3% 21: 8 – 2 – 2 – 1 1,221,496,848 0.2%
22: 7 – 3 – 2 – 1 11,943,524,736 1.9% 22: 8 – 3 – 1 – 1 746,470,296 0.1%
23: 7 – 2 – 2 – 2 3,257,324,928 0.5% 23: 8 – 3 – 2 – 0 689,049,504 0.1%
24: 6 – 6 – 1 – 0 459,366,336 0.1% 24: 7 – 5 – 1 – 0 689,049,504 0.1%
25: 6 – 5 – 2 – 0 4,134,297,024 0.7% 25: 6 – 6 – 1 – 0 459,366,336 0.1%
26: 6 – 5 – 1 – 1 4,478,821,776 0.7% 26: 8 – 4 – 1 – 0 287,103,960 <0.1%
27: 6 – 4 – 3 – 0 8,421,716,160 1.3% 27: 9 – 2 – 1 – 1 113,101,560 <0.1%
28: 6 – 4 – 2 – 1 29,858,811,840 4.7% 28: 9 – 3 – 1 – 0 63,800,880 <0.1%
29: 6 – 3 – 3 – 1 21,896,462,016 3.4% 29: 9 – 2 – 2 – 0 52,200,720 <0.1%
30: 6 – 3 – 2 – 2 35,830,574,208 5.6% 30: 7 – 6 – 0 – 0 35,335,872 <0.1%
31: 5 – 5 – 3 – 0 5,684,658,408 0.9% 31: 8 – 5 – 0 – 0 19,876,428 <0.1%
32: 5 – 5 – 2 – 1 20,154,697,992 3.2% 32: 10 – 2 – 1 – 0 6,960,096 <0.1%
33: 5 – 4 – 4 – 0 7,895,358,900 1.2% 33: 9 – 4 – 0 – 0 6,134,700 <0.1%
34: 5 – 4 – 3 – 1 82,111,732,560 12.9% 34: 10 – 1 – 1 – 1 2,513,368 <0.1%
35: 5 – 4 – 2 – 2 67,182,326,640 10.6% 35; 10 – 3 – 0 – 0 981,552 <0.1%
36: 5 – 3 – 3 – 2 98,534,079,072 15.5% 36: 11 – 1 – 1 – 0 158,184 <0.1%
37: 4 – 4 – 4 – 1 19,007,345,500 3.0% 37: 11 – 2 – 0 – 0 73,008 <0.1%
38: 4 – 4 – 3 – 2 136,852,887,600 21.6% 38: 12 – 1 – 0 – 0 2,028 <0.1%
39: 4 – 3 – 3 – 3 66,905,856,160 10.5% 39: 13 – 0 – 0 – 0 4 <0.1%
Total 635,013,559,600 Total 635,013,559,600

Source: Durango Bill’s

But don’t be overwhelmed ! Even from all 39, we can manage them much better if we treat them as 5 separate categories. In order of less frequent to the most frequent, they are:

 

Distribution Total Hands Probability
1: 13 – 0 – 0 – 0 4 <0.1%
2: 12 – 1 – 0 – 0 2,028 <0.1%
3: 11 – 2 – 0 – 0 73,008 <0.1%
4: 11 – 1 – 1 – 0 158,184 <0.1%
5: 10 – 3 – 0 – 0 981,552 <0.1%
6: 10 – 2 – 1 – 0 6,960,096 <0.1%
7: 10 – 1 – 1 – 1 2,513,368 <0.1%
8: 9 – 3 – 1 – 0 63,800,880 <0.1%
9: 9 – 2 – 2 – 0 52,200,720 <0.1%
10: 9 – 2 – 1 – 1 113,101,560 <0.1%
0.04%

Extreme 1 Suiter Hand

With total of less than 0.04% of distribution (Roughly you can imagine this by playing 2500 boards of Bridge and only found 1 of them only once), these 10 type of hand distribution is relatively easy to bid. Just bid the suit at 4, 5 or even 6 or 7 level and done.

Well, if the HCP count is normal opening, you might want to open as “normal” as partner might have some more point too .

For all other 1 suiter with 8 cards and below, I put them on separate category that statistically will be more often to be found in the real play, therefore more wok need to be focused on rather than this extreme category

3 Suiters hand

 

Distribution Total Hands Probability
1: 5 – 4 – 4 – 0 7,895,358,900 1.2%
2: 4 – 4 – 4 – 1 19,007,345,500 3.0%
4.24%

Three suiters hands are quite notoriously difficult to bid/rebid during “normal” bidding and sometimes miss the fir on the third suits. There fore some bidding system is actually treating this distribution seriously and creating a special opening bid or seqence just to cater for this bid. But luckily it should not come very often as the probability of all 3 suiters types are just above 4% (meaning for every 25 boards you will encounter one)

1 Suiter hand

 

Distribution Total Hands Probability
1: 8 – 3 – 2 – 0 689,049,504 0.1%
2: 8 – 3 – 1 – 1 746,470,296 0.1%
3: 8 – 2 – 2 – 1 1,221,496,848 0.2%
4: 7 – 3 – 3 – 0 1,684,343,232 0.3%
5: 7 – 3 – 2 – 1 11,943,524,736 1.9%
6: 7 – 2 – 2 – 2 3,257,324,928 0.5%
7: 6 – 3 – 3 – 1 21,896,462,016 3.4%
8: 6 – 3 – 2 – 2 35,830,574,208 5.6%
12.17%

I combine 6,7 and 8 cards holding into 1 category as this is more often than the rest (9+ card 1 suiter)

Your system need to cater the distribution on this group as it represent 12.17% or 1 board for every 8 played.

The common approach will be:

  • 7 and 8 cards weak, open at level 3
  • 6 card Major weak, open at level 2
  • Others opening level 1 as normal

2 Suiters Hand

 

Distribution Total Hands Probability
1: 9 – 4 – 0 – 0 6,134,700 <0.1%
2: 8 – 5 – 0 – 0 19,876,428 <0.1%
3: 8 – 4 – 1 – 0 287,103,960 <0.1%
4: 7 – 6 – 0 – 0 35,335,872 <0.1%
5: 7 – 5 – 1 – 0 689,049,504 0.1%
6: 7 – 4 – 2 – 0 2,296,831,680 0.4%
7: 7 – 4 – 1 – 1 2,488,234,320 0.4%
8: 6 – 6 – 1 – 0 459,366,336 0.1%
9: 6 – 5 – 2 – 0 4,134,297,024 0.7%
10: 6 – 5 – 1 – 1 4,478,821,776 0.7%
11: 6 – 4 – 3 – 0 8,421,716,160 1.3%
12: 6 – 4 – 2 – 1 29,858,811,840 4.7%
13: 5 – 5 – 3 – 0 5,684,658,408 0.9%
14: 5 – 5 – 2 – 1 20,154,697,992 3.2%
15: 5 – 4 – 3 – 1 82,111,732,560 12.9%
16: 5 – 4 – 2 – 2 67,182,326,640 10.6%
35.90%

Any distribution with 5+cards in 1 suit and 4+ cards in another suit, I consider them as second suit.

This group is responsible for more than 1/3 of all hand that you play. Hence, a careful bidding as how to rebid both suits is very important to make sure a fit in second suit will always be found without confusing it with point level. What I mean is say you have 5 cards and 5 cards , then you need to bid 1 first and rebid 2, whilst if you have very strong hand (e.g: 19 HCP) you can bid 1C first and reverse bid to 2.

Balanced Hand

 

Distribution Total Hands Probability
1: 5 – 3 – 3 – 2 98,534,079,072 15.5%
2: 4 – 4 – 3 – 2 136,852,887,600 21.6%
3: 4 – 3 – 3 – 3 66,905,856,160 10.5%
47.6%

You probably argue that 5-3-3-2 is not balanced. Yes, it’s actually semi balanced. But most of the system is now treat 5-3-3-2 as balanced as you cannot really re-bid with other suit. Rebid the same suit will promised additional card or minimum, so treating it as balanced similar to 4-3-3-3 is probably a good idea.

This category is the biggest group. 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 bridge learning. 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 with balanced hand sequence first !

Warning: Computer Dealt hand

All these distributions that we are talking above are based on normal distribution. But a note need to be put here that with more and more distribution dealt by computer (where the administrator can skew the distribution to make it more interesting for certain tournament, i.e: make sure some exotic grand slam is dealt,etc) – So, whilst more work still need to be focused on highly probable ones, your system still need to include all the possibilities.

5 comments

  1. greg /

    What is the probability of being dealt a bridge hand with no card higher than an 8??

  2. Colin Aldridge /

    How do you calculate hand distributions of a single suit. For example I hold AKQxxx of a suit. What are the chances that the suit will break 322,331,421,430,511,520,610,700

  3. Sang /

    Hey, I’m doing a maths project at school about bridge relating it to probability.

    I’m just wondering how you worked out the probability of attaining each hand?
    It would be most appreciated if you could email me the method/technique used.

    Kind regards

    Sang

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