As
you know by now, there are 39 different suit-distributions
in a bridge hand – the so called “shapes”.
Here
they are:
ALL 39 possible distribution
patterns
4-3-3-3
4-4-3-2
4-4-4-1
5-3-3-2
5-4-2-2
5-4-3-1
5-4-4-0
5-5-2-1
5-5-3-0
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
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
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
A
natural bridge-question is what would be the probability
of these shapes to “land” in your hand.
The
table below covers the probability of their occurrence:
Hand Distributions with their
Probabilities
4-3-3-3
= 10.5%
4-4-3-2
= 21.5%
4-4-4-1
= 3.0%
5-3-3-2
= 15.5%
5-4-2-2
= 10.5%
5-4-3-1
= 13.0%
5-4-4-0
= 1.3%
5-5-2-1
= 3.2%
5-5-3-0
= 0.9%
6-3-2-2
= 5.6%
6-3-3-1
= 3.5%
6-4-2-1
= 4.7%
6-4-3-0
= 1.3%
6-5-1-1
= 0.7%
6-5-2-0
= 0.6%
6-6-1-0
= 0.1%
7-2-2-2
= 0.51%
7-3-2-1
= 1.88%
7-3-3-0
= 0.26%
7-4-1-1
= 0.39%
7-4-2-0
= 0.36%
7-5-1-0
= 0.10%
7-6-0-0
= ~0
8-2-2-1
= 0.19%
8-3-1-1
= 0.12%
8-3-2-0
= 0.10%
8-4-1-0
= ~0
8-5-0-0
= ~0
9-2-1-1
= 0.02%
9-2-2-0
= 0.01%
9-3-1-0
= 0.01%
9-4-0-0
= ~0
10-1-1-1
= ~0
10-2-1-0
= ~0
10-3-0-0
= ~0
11-1-1-0
= ~0
11-2-0-0
= ~0
12-1-0-0
= ~0
13-0-0-0
= ~0
The
numbers marked as ~0 are numbers less than 0.01%. It
is worth noticing that the 4-3-3-3 distribution is not
among the top 3 most probable distributions and that
by far the most probable one is 4-4-3-2 – 6% above the
second-most-probable 5-3-3-2.
The
distributive part of the Zar Points varies from 8
for flat hand to 26 for the “wildest” hand with
3 voids. This means that it classifies the hands
in 17 categories.
Here
they go:
Zar Distribution Points for
ALL distributions
4-3-3-3
= 8
4-4-3-2
= 10
4-4-4-1
= 11
5-3-3-2
= 11
5-4-2-2
= 12
5-4-3-1
= 13
5-4-4-0
= 14
5-5-2-1
= 14
5-5-3-0
= 15
6-3-2-2
= 13
6-3-3-1
= 14
6-4-2-1
= 15
6-4-3-0
= 16
6-5-1-1
= 16
6-5-2-0
= 17
6-6-1-0
= 18
7-2-2-2
= 14
7-3-2-1
= 16
7-3-3-0
= 17
7-4-1-1
= 17
7-4-2-0
= 18
7-5-1-0
= 19
7-6-0-0
= 20
8-2-2-1
= 17
8-3-1-1
= 18
8-3-2-0
= 19
8-4-1-0
= 20
8-5-0-0
= 21
9-2-1-1
= 19
9-2-2-0
= 20
9-3-1-0
= 21
9-4-0-0
= 22
10-1-1-1
= 20
10-2-1-0
= 22
10-3-0-0
= 23
11-1-1-0
= 23
11-2-0-0
= 24
12-1-0-0
= 25
13-0-0-0
= 26
Finally,
we’ll present the table handling all these 3 simultaneously
– the shapes, their probability, and the Zar Points
associated with any of them:
Hand Distributions with their
Probabilities and Zar Points assigned
4-3-3-3
= 10.5% 8 ZP
4-4-3-2
= 21.5% 10 ZP
4-4-4-1
= 3.0% 11 ZP
5-3-3-2
= 15.5% 11 ZP
5-4-2-2
= 10.5% 12 ZP
5-4-3-1
= 13.0% 13 ZP
5-4-4-0
= 1.3% 14 ZP
5-5-2-1
= 3.2% 14 ZP
5-5-3-0
= 0.9% 15 ZP
6-3-2-2
= 5.6% 13 ZP
6-3-3-1
= 3.5% 14 ZP
6-4-2-1
= 4.7% 15 ZP
6-4-3-0
= 1.3% 16ZP
6-5-1-1
= 0.7% 16 ZP
6-5-2-0
= 0.6% 17 ZP
6-6-1-0
= 0.1% 18 ZP
7-2-2-2
= 0.51% 14 ZP
7-3-2-1
= 1.88% 16 ZP
7-3-3-0
= 0.26% 17 ZP
7-4-1-1
= 0.39% 17 ZP
7-4-2-0
= 0.36% 18 ZP
7-5-1-0
= 0.10% 19 ZP
7-6-0-0
= 0.006 20 ZP
8-2-2-1
= 0.19% 17 ZP
8-3-1-1
= 0.12% 18 ZP
8-3-2-0
= 0.10% 19 ZP
8-4-1-0
= 0.045 20 ZP
8-5-0-0
= 0.003 21 ZP
9-2-1-1
= 0.02% 19 ZP
9-2-2-0
= 0.01% 20 ZP
9-3-1-0
= 0.01% 21 ZP
9-4-0-0
= 0.001 22 ZP
10-1-1-1
= 0.0004, 20ZP
10-2-1-0
= 0.0011, 22ZP
10-3-0-0
= 0.0002, 23ZP
11-1-1-0
= ~0
11-2-0-0
= ~0
12-1-0-0
= ~0
13-0-0-0
= ~0
The
three methods discussed in the article for comparative
reasons assign different weights to these 39 shapes
and we present how these 4 methods do it in the table
below.
The
table is ordered by the amount of Zar Points assigned,
in ascending order.
As
might be expected, ALL methods basically follow the
same ascending line, giving the least amount of points
for the balanced distributions and the biggest amount
of points for the “wildest” distributions.
Since
for everyone the 4-3-3-3 case is the “base”
to which everybody assigns the minimum points we are
going to consider only the rest of the groups in the
evaluation methods (taking 4333 distribution as base).
In
the table below, the columns of the table are the displacements
from the “base”, (e.g. +1 means the first group after
the base of 4-3-3-3) while the actual number in the
body of the table represent the number of distributions
the corresponding group.
Marty
Bergen’s Points classifies the hands in 6 groups,
the 3-2-1in 9, Drabble in 5, and Zar Points in
17. This means by the criteria of span of base
(number of classification groups) Zar points are between
2 to 3.4 times better than the rest of the methods.
The
separation power of the methods is given by the
max number of distributions in a group. In Zar Points
this number is 4, while Bergen has 9, Goren – 8, and Drabble – 13. Again - between
2 and 3.2 times better results.
When
we take into account the number of elements (hands)
in each group, we can now find the Standard Deviation
for each method and see the difference there. Here is
what is meant by that.
The
root-mean-square (RMS) of a variant x, sometimes called
the quadratic mean, is the square root of the mean squared
value of x:
Scientists
often use the term root-mean-square as a synonym
for standard deviation when they refer to the square
root of the mean squared deviation of a signal from
a given baseline or fit.
Applying
the standard deviation from the basis (the x coordinate)
measure to the three hand-evaluation methods (using
the number of hands in each group) yields the following:
So
by this 3rdcriteria, the standard deviation
of the evaluation method, Zar Points demonstrate between
2.2 and 3.6 times better results.
The
interesting part is that by ANY of the applied
three criteria:
1)
Span of base
2)
Separation power
3)
Standard Deviation
Zar
Points manifests roughly threetimes better
results than any of the threecompetitors.
Let’s
have a look at the WBF definition of an opening hand.
Much
to your surprise, I guess, the WBF doesn’t define an
opening hand in terms of Goren Points, or Bergen Points,
or (nota bene) even in terms of Milton Work HCP!
WFB was smart enough not to engage with embracing any
point count
The
rule actually says that an opening hand is a hand “better
than the average hand with a Queen worth!
This
means that basically regardless of how you measure
the hands, an opening hand is a Queen better than the
average – so you only have to calculate the worth of
the average hand in your method and then add
a Queen-worth to see if your method complies with the
WBF standard for opening.
For
example, and average HCP-measured hand is a hand with
10 HCP points. Hence, an opening hand in Milton Work
sense is a hand with 10 + 2 = 12 HCP.
So,
let’s measure the average Zar Points hand.
The
average hand in terms of 6-4-2-1 Honor Points contains
6+4+2+1 = 13 points. Now, for the Distributive
Portion, we have to go to the each distribution and
multiply the Zar Points it brings with the probability
this distribution to occur. Then, when we add-up all
the results, we’ll come up with the average Zar Points
distributional hand.
