We will assess the accuracy of four different methods of bridge distribution
evaluation via some standard common
mathematical approaches.
The first one is the already mentioned CharlesGoren’s system, known as the “3-2-1” system, named after
the points assigned for short-suits
holdings.
The second method is the MartyBergen’s “Rule of 20” method
from his famous book-series “Points
Schmoints”.
The approach of Bergen is to assign points equal
to the sum of the lengths of the 2 longest
suits of a hand, i.e. (a+b), using our notation.
We will also compare with the newest method from the
late 90-ies, the Drabble
rule of “adding the 2 longest
suits, divide by 3, and subtract the
length of the shortest suit, rounding
downwards. Since Drabble’s
scale starts with -1 for the 4-3-3-3,
we have adjusted it by shifting the
entire table with (+1) to eliminate
the negative numbers.
In all cases we consider the initial base points, before
the “fine tuning” in one
way or another.
The fourth method is the Zar distribution Points method you are already familiar with - assigning
the value of (a+b)
+ (a-d), i.e. the sum of your 2 longest
suits, plus the difference between your
longest and your shortest suit (effectively
representing the SUM of all the 3 suit-differences
of the hand).
As
we mentioned, there are 39 different
suit-distributions in a bridge hand.
The
table below covers them, along with
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
We are going to compare the 4 methods by 3 criteria:
1)span of base, given by the number of the groups the method classifies the hands in;
2)separation power, given by the maximum number of distributions which can fall in a single
group;
3)standard deviation, which is explained below
in the article.
To prepare for this exercise, we will present the following
table with the points assigned by all
four evaluation methods:
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:
