Passion of The Game of Bridge

Work Out The Percentage or Probability of (Opening) Hand Distribution During Bidding

Perhaps knowing the statistic of every possible hand only need to be known by Expert that develop a bidding system. But sometimes, you just need to know which is the most probable hand that your partner has. Why? Of course to give you the best contract and giving you the top board !

For example: partner open 1S (natural) and you only have doubleton Spade, 3 cards Heart, 4 cards Diamond 4 cards Club with 8HCP. Playing 1NT Forcing, you just bid 1NT in which partner rebid with 2C (better minor, could be 3 card –  only when 5-3-3-2). The question: should you play 2C (with possibility 4-3 fit) or play 2S(with at least 5-2 fit)? Well, 2S could be 1 down while 2C is making -or- 2S got 8 tricks but 2C have 9 tricks -or- the other way around. Of course many other factors such as how the defense conducted, the location of high card, the distribution of the card, etc play a big role making it impossible to predict each individual case.

So, this is where the statistic is useful. If you follow the probability number, in the long run you will get better result.

Hand Distribution Probability

Let us starts with the hand distribution probability below (simplified, distribution not mentioned below have very very small probability – see this for more detail):

Hand Distribution Probability
Distribution (%) Distribution (%)
4-4-3-2 22 6-5-2-0 0.7
4-3-3-3 11 6-6-1-0 <0.1
4-4-4-1 3 7-3-2-1 2
5-3-3-2 16 7-2-2-2 0.5
5-4-3-1 13 7-4-1-1 0.4
5-4-2-2 11 7-4-2-0 0.4
5-5-2-1 3 7-3-3-0 0.3
5-4-4-0 1.2 7-5-1-0 0.1
5-5-3-0 0.9 7-6-0-0 <0.1
6-3-2-2 6 8-2-2-1 0.2
6-4-2-1 5 8-3-1-1 0.1
6-3-3-1 3 8-4-1-0 <0.1
6-4-3-0 1.3 8-5-0-0 <0.1
6-5-1-1 0.7 9-2-1-1 <0.1

With above example, without being too technical about statistic, we can say that the probability partner has 3 card C is only if the distribution is 5-3-3-2 (16%). While the possibility partner has 4 or more card is when the distribution is 5431 or 5422 or 5440 (13+11+1.2=25.2% probability) – of course assuming with 5-5 partner will bid 3C and with 6+card S partner would bid 2S. Therefore we can conclude that possibility partner has 4 card is higher than 3 card. So in above example, you should always pass and hope for the best.

Warning: to be statistically correct, we just can say the relativity over another (25.2% is higher than 16%) but the actual number of the probability (25.2% and 16%) is not really accurate as we need to consider that the 5 cards could be from 3 other suits as well (i.e: considering there are 4 suits that can become the 5 cards, then the probability is closer to 16%/4 = 4% from total population, etc).

High Card Distribution

The 2nd useful table is the high card distribution probability (see below)

High Card Distribution
HCP (%) HCP (%)
0 0.4 16 3.3
1 0.8 17 2.4
2 1.4 18 1.6
3 2.5 19 1.0
4 3.9 20 0.6
5 5.2 21 0.4
6 6.6 22 0.2
7 8.0 23 0.1
8 8.9 24 0.06
9 9.40 25 0.03
10 9.41 26 0.01
11 8.9 27 0.005
12 8 28 0.002
13 7 29 0.001
14 5.7 30 0.0002
15 4.4 31-37 0.0001

As you can see the probability of having 10 HCP is the highest (40 total HCP divided to 4 person) and gradually decreasing toward 0 and 40.

So, what is the probability partner has 12-21 HCP? Easy, just add all the percentage: 8+7+5.7+4.4+3.3+2.4+1.6+1.0+0.6+0.4 = 34.4%

Then how much probability partner open 1H (5+ card H 12-21)?
Answer: 34.4% (From point range: 12-21) x 69% (all hand distribution with 5,6,7,8,9 card) x 25% (1 out of 4 suit) = 6% (1 in 17 board)

How much the probability partner open 2H (weak 2, 6-10HCP, 6 card H)?
42.3% (from point/HCP %: 6.6+8+8.9+9.4+9.4) x 15.3 (from distribution: 6-3-3-2, 6-4-3-1, 6-4-2-1, 6-4-3-0) x 25% (1 out of 4 suit) = 1.6%.

So, in this case opening 1H has about 4x more probability) than 2H.

Therefore with similar method we can construct a popular opening hand possibility as follow:

Opening Probability Relative Description
1m 8.60% 18 12-21 hcp, at least 3cd m
1M 6.00% 12 12-21 hcp, 5cd M
1NT 4.95% 10 15-17 hcp: 5332, 4432, 4333
2C 0.81% 2 Strong 20+
2D/2H/2S 1.60% 3 Weak 2, 6-10 hcp
2NT 0.49% 1 BAL 20-21
3X 0.49% 1 Preemptive at least 7-8cd X – 5-10
47

The above table means for ever 2NT opening that you did, you, on average, will open 10 times 1NT, 18 times 1C or 1D, about 2 times 2C, etc.

This table tells you that if you just learning a new system, start with 1m, 1M and 1NT to cover the most wide spectrum hand that you can handle. Do not worry too much about 2NT or 3X sequence as they will rarely come.

Hope this can be useful.

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