Statistical Analysis of Corrections on LoTT
[0]
Outline
Corrections (or adjustments) appear to be of utmost concern in the topic of
LoTT (the Law of Total Tricks). In this article, results are
quoted from the newest version of my freeware
LottAnalyzer, which are
based on Ginsberg's doubledummy library of 717,102 random deals.
They mostly agree with recommendations of forerunners, but disagree in detail.
The corrections have usually been discussed as to if they are justifiable.
In the present statistical analysis, frequencies (probabilities) will also be studied for
various types of extra factors (other than total trumps) that will require corrections.
« Frequencies of
Major Extra Factors »
Extra Factors  Frequency

Double Double Fit  12.0%

Double Solid Trumps  8.5%

Nortrump vs Singleton  47.3%

Naturally, those extra factors which appear more often would need be more closely studied.
For example, let me extract this table from the result below, where all three
factors deserve a full +1 trick correction.
In this table, the first two lines refrer to the suitvssuit contracts, while the
bottom line to Notrumpvssuit contracts.
[2]
Method
When we speak of such and such corrections to LoTT, oddly enough, such and such
cases will actually take place rather infrequently.
Thus, if we take statistics over the whole ensemble of random deals, as has
been initiated on a large scale by Matthew Ginsberg and extended by myself, such
and such cases are smeared out, and we are unable to draw any conclusions
on corrections from statistical analysis.
To avoid this, a new feature has been introduced into the above software,
which will take statistics over only limited class of deals which satisfy such
an such conditions.
[3]
Result
At present, the software allows some 15 classes, the result from which are
some clear and some unclear. The clearer ones will be mentioned below.
The software allows, in addition, to specify the range of HCPs.
This restriction on HCPs will not be placed below, because the results have been found to
not depend on HCPs,
(although often claimed to the
contrary without definite proof).
Furthermore, there remains a choice between the "Length Rule" and
"Score Rule" on how to select trump suits. The former
rule
(chosen by Ginsberg, with Notrump calls totally ignored) takes automatically
the longest suit as the trump, as if thinking it best, while the latter selects the trump suit
(or Notrump) so that the net score be biggest for both sides, even in the presence of
bitter competition and possible sacrifice bids.
In the analysis below, the latter is adopted, since it is more natural
at bridge table. With this choice, there remain no ambiguities on selection of
trump suits.
[3a]
Double Fit
As usual, "Double Fit" is understood to mean 8 (+) cards in two suits in
two hands. Let us begin with this class.
Is or isn't "Double Fit" a positive extra factor
by itself ?
« Probability of Double Fits
in "Suit vs Suit" Deals »
Owner  Probability

Both have  12.0%

Only one side has  24.2%

Neither has  63.8%

As a matter of fact, Jean René Vernes denies, and Larry Cohen emphatically
denies. They agree on a positive correction only when
the "Double Fit" is Double. That is to say, they count only
when Both sides have a double fit.
We first ask how often a doubledouble fit will take place.
The answer is given in this table, and we know that a doubledouble fit
occurs with probability 12.0%, surely infrequently, almost once in 8 deals, except
NotrumpvsSuit deals.
Next we restrict ourselves to the "doubledouble fit" class; we
then obtain a figure like this, which consists of three parts:
 The bar graphs (in light blue, scale left) show the frequency of
various total trumps.
Here, total trumps 16 and 18 will both take place with high probability.
The singular behavior of this frequency distribution will be
separately explained below (actually, the above emphasis will turn out to
be unnecessary).
 The red crosses (scale right) show averages of actual total tricks
with the error bars.
Their length means that the average error is about 1 trick.
[When averaged over the total trumps, the deviation from the LoTT [A]
turns out to be +0.92 tricks with an average error of 0.87 tricks.]
 The blue solid line with inclination 45°represents the LoTT line
[A] Total Tricks = Total Trumps,
whereas the dotted line
[B] Total Tricks = Total Trumps + 1
has been drawn for guide to eyes.
From this result, we observe that +1 trick correction is quite appropriate
up to 19 total trumps. Simple double fit is not enough for +1 trick
correction
; only a doubledouble fit is required.
You may then ask what in case of a simple double fit (24.2%)
.
The result is as you may easily imagine
: perhaps, +0.5 trick correction ?
(
click here).
[3b]
Solid Trumps
Under this class, the software assumes the presence of
A,
K, and
Q
in the trump suit in combined hands.
Should this holding
AKQ be Double, too ?
« Probability of
Solid AKQ trumps »
Owner  Probability

Both sides have  8.5%

Only one side has  36.0%

Neither side has  55.5%

This time, we hear no answers from both authors.
They both claim importance of solid trumps, but neither clarifies if
Simple solid trumps will be enough, or Double is necessary.
So, we try.
We start again to see how single/double solid trumps are likely
.
The table shows that Double solid trumps occur with a small probability 8
.5%, again
infrequently.
We then take statistics over these only 8.5% cases.
The resulting figure is similar to the above doublefit case.
Only with doublesolid trumps, +1 trick correction is to be counted, up to
18 Total Trumps.
From 19 total trumps on, the tricks grow in less proportion to total trumps.
This convex behavior of total tricks is a common feature
: LoTT will tend to
overestimate
tricks for increasing trumps.
"Too many trumps" is a negative factor to LoTT
(Just consider if you can hold down 2 losers, simply because you have 11 trumps).
You may then ask what in case of a simple solid trump.
Perhaps you can easily imagine, but
click here and see.
[3c]
Chameleons
I love the short term "Chameleons" for minor honors in opponents'
trump suit. They will sometimes win a trick in defense, but are useless in
their own contract, just as chameleons behave differently under
circumstances.
Even though there can be many kinds of chameleons, the software assumes, under
this class, only the threecard holding
Qxx, where
x may include J and 10.
« Frequency of Chameleon Qxx »
 Owner  Frequency

(a)
 Both have  0.8%

(b)
 Only one side has  16.4%

(c)
 Neither side has  82.8%

We have to note first how often chameleons will appear.
The frequency values in the table were obtained from random 480 thousand deals
(except Notrump deals).
So, we observe again that the phenomenon is infrequent.
In particular, it seldom appears in both sides together, and will never appear in
82.8% deals.
« Average Total Tricks
With and Without Chameleons »
Total Trumps  Both have  Only one side has  Neither has

15  0.37±0.64  0.51±0.70  0.66±0.71

16
 0.11±0.68  0.36±0.76  0.59±0.79

17
 0.16±0.79  0.30±0.82  0.48±0.85

18
 0.27±0.87  0.25±0.87  0.33±0.90

19
 0.40±0.89  0.20±0.88  0.04±0.87

20
  0.10±0.89  −0.27±0.95

average
 0.25±0.73  0.37±0.78  0.44±0.82

Average tricks obtained in these three cases do not differ so much as to
allow clear graphical comparison.
Hence, we show the result in this table
:
Up to 18 total trumps, absence of chameleons is more a positive factor than
its presence is a negative factor. Magnitude of correction is on the
order of ¼ tricks.
Nevertheless, this favorable trend changes for larger total trumps for some reason unknown to me.
Advantage of absent chameleons appears to be effective only for smaller
total trumps.
Certainly, when a deal has lots of trumps and hence lots of tricks, other
factors will dominate over presence/absence of chameleons.
[3d]
Notrump vs Singleton
We next consider deals where one side is playing Notrump
.
Correction (or adjustment, or LoTT formula itself) proposed by
JeanRené Vernes reads
[V] Total Tricks = Number of Trumps + 8, 7, 6,
where the three numbers on the right correspond to a void, singleton and doubleton
in a side suit of opponents.
As to the owner of this void/singleton, we
follow JeanRené Vernes to count a void/singleton in (supposed) dummy as well as
in (supposed) declarer.
We have also Larry Cohen's version in
To Bid or Not To Bid:
[C] Total Tricks = Number of Trumps + 8, 7½, 7.
« Frequency Distribution of Opponents' Shape »
Shape  Frequency

Void  9.6%

Singleton  47.3%

Doubleton  43.1%

Let us begin with relative frequency of a void, singleton, and doubleton.
The table on the right has been obtained from 236 thousand random deals (except
suitsuit contracts).
Please notice that this frequency distribution is
appropriate only when one side is going to play Notrump.
Rather flat hands of Notrumpers will tend to make the opponents' hands also
flat.
As a result, voids will appear more infrequently, but singletons
remain still frequent.
We now ask
LottAnalyzer
for the Singleton case, to see which formula is better, [V] or [C] ?
In this graph, two lines are drawn for guide to eyes, the solid line for
Number of trumps + 7, and the broken line for Number of trumps + 8.
This graph clearly shows that, in the "Notrump vs Singleton" class,
neither [V] nor [C] works better than the proposed Singleton formula,
[S] Total Tricks = Number of Trumps + 8 (for singletons).
You may ask what if opponents have only doubletons,
just click here. You will find for yourself a Notrump formula
for doubletons.
As for voids, the tricks heavily depend on the number of trumps, and
we are unable to draw such a simple formula for voids.
The tricks available will depend more on bidding situations, e.g., whether
you are overcalling against NT opening in the direct position or in the
balancing position.
[4]
DoubleFit Rule
In this paragraph, we look back to our
first figure (reproduced
on the right) and consider the origin of its singular distribution.
To set up a handy notation, let [8d] denote 8+8 double
fits (8 cards in two suits), and let [9d] denote 9+8
as well as 9+9 double fits.
These two kinds of double fits will take place in 10.34% and 7.88% deals,
respectively. We have discussed this kind of probability
elsewhere(Sec.6) in depth,
but we will repeat in the present context.
Now, it is obvious that the double peaks at 16 and 18 trumps arise, respectively,
from the [8d]+[8d] doubledouble fits and the [9d]+[9d]
doubledouble fits. In this connection, it is interesting
to note that the [8d] and [9d] double fits
can never coexist.
« Correlation between Our and Their Shapes When We Have an [8d] (10.34%) »
 Our Shape  Their Shape  Double Fit  Frequency

(a)  [8,8,8,2]  [5,5,5,11]  none  × 1.3%

(b)  [8,8,7,3]  [5,5,6,10]  none  ×19.0%

(c)  [8,8,6,4]  [5,5,7,9]  none  ×47.6%

(d)  [8,8,5,5]  [5,5,8,8]  *  ×32.1%

To put the other way, suppose your side have an [8d] double fit, then
they (opponents) can certainly have an [8d] double fit, but never can they
have a [9d] double fit.
This is an aspect of the "DoubleFit Rule", which may be
proved plainly
:
When you have an [8d] double fit, your shape should be one of (a) −
(d) listed in the table, and correspondingly, opponents will have the
complementary shape of 13.
Their possible double fit is hence limited to only [8d] realized in the case (d).
Their double fit can never be [9d].
« Correlation between Our and Their Shapes When We Have a [9+8] Double Fit (6.92%) »
 Our Shape  Their Shape  Double Fit  Frequency

(a)  [9,8,8,1]  [4,5,5,12]  none  × 0.5%

(b)  [9,8,7,2]  [4,5,6,11]  none  × 8.6%

(c)  [9,8,6,3]  [4,5,7,10]  none  ×31.6%

(d)  [9,8,5,4]  [4,5,8,9]  *  ×59.3%


Correlation When We Have a [9+9] Double Fit (0.97%)
 Our Shape  Their Shape  Double Fit  Frequency

(a)  [9,9,8,0]  [4,4,5,13]  none  × 0.2%

(b)  [9,9,7,1]  [4,4,6,12]  none  × 2.9%

(c)  [9,9,6,2]  [4,4,7,11]  none  ×17.1%

(d)  [9,9,5,3]  [4,4,8,10]  *  ×47.1%

(e)  [9,9,4,4]  [4,4,9,9]  *  ×32.7%


Its extension to the 9+8 double fit is straightforward
:
If you have a 9+8 double fit, their double fit is limited also to a 9+8 double fit,
(again the bottom case (d) in the list).
In case (though very unlikely) we have a 9+9 double fit, they can have two kinds of
double fits (d) and (e), which can never be 9+8 nor 8+8. Q.E.D.
Thus, we have shown that the three double fits 8+8, 9+8, and 9+9 have
no intersections each other.
We note in passing that your [9d] has more chance
(than 32.1% in [8d]) for
opponents to have a double fit, too.
This fact is perhaps, at least qualitatively, known to many bridge players.
The table below shows
quantitatively that, the more shapely hands you two
have got, the more shapely hands opponents are likely to have.
The difference between the [8d] and [9d] double fits is really great.
Probability of Double and DoubleDouble Fits
Your Shape  Double Fit  Double Double Fit

8d  10.34%  ×32.11%

98  6.92%  ×59.25%

99  0.97%  ×79.84%

10d  1.74%  ×82.53%

11d  0.20%  ×93.77%

From this doublefit rule, we easily understand why the distribution is strongly
peaked at 8+8=16 and 9+9=18 trumps.
It's OK for 16 and 18, but what about 17 trumps ?
If the [8d] and [9d] double fits really never coexist, frequency of
the 17 trumps should be strictly zero.
Why is it nonvanishing at 17 trumps in that graph ?
Actually, there is no contradiction.
If we select automatically the longest suit as the trump suit, then entry in the 17
trumps will be strictly zero, indeed
.
But, bridge bidding is not so simple.
Suppose that you have a 9+8
double fit, which consists of
44 cards in
and 54 in
,
while opponents are most likely (59.3%) to have also a 9+8 double fit, 5+4 in
and 4+4 in
.
Against their
3 bid, you will compete with 3
, with shorter
than
.
As a result, total trumps will become 8 (your
) + 9 (their
) = 17.
The doublefit rule is very useful at bridge table. Owing to this rule,
you don't need to worry about if the deal is doubledouble or singledouble.
The logic is easy:
When you have an [8d] fit, the doublefit rule will guarantee that opponents will
have
at least an [8d] fit.
In that case (32.1%), you can expect
8 + 8 + 1 correction = 17 total tricks,
because of the doubledouble fit
;
or else they have at least 9+7 shape (47.6%) or more, and hence
8 + 9 = 17 tricks (from the LoTT formula).
In either case, you can expect 17 tricks. Therefore, it is unnecessary to
emphasize
doubledouble fit. The deal is effectively
doubledouble, once you have a double fit. So, it suffices to say simply
double fit.
[5]
Postscript
We have so far examined various possible corrections to LoTT, together with
the probabilities with which they take place.
We have found most of them are rather infrequent.
Extra Factors for Corrections  Frequency

Double Double Fit  12.0%

Double Solid Trumps  8.5%

Nortrump vs Singleton  47.3%

So, it is not clever, particularly for beginners, to be overly sensitive to
these corrections. An important exception is the "Notrump vs Singleton"
case, which will occur with probability 47.3%, and the above statistical analysis
strongly suggests the proposed LoTT formula [S] for that case.
In case you are anxious about detailed corrections on LoTT, the
earlier remark (p.40)
by JeanRené Vernes is to be remembered. He compares results
of all the deals in 19531959 world championships, which were played in
two rooms
with the same trump, on the same level, and played by the
same side. The number of tricks taken varied ±0.52 tricks, even
among this level of players;
from which [regarding it as an unavoidable error,] he concludes that a formula which could predict the number of total
tricks with an accuracy of a half trick may be considered as rigorously perfect.