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Law of Total Tricks - a parity oddity



I was using my program, Deal, to do some experiments pertaining to The Law,

when I came across something that took me a bit of time to diagnose.



Larry Cohen says to add an adjustment of one trick to the total tricks if

both sides have two 8+-card fits. I decided to determine how often such

double-double fits (DDFs) occur.



How often does a double-double fit occur when the total trumps is n, for

n=14...23?



The table is surprising:



Total trumps   % with DDFits

14               0.0%

15               0.0%

16              11.9%

17               0.0%

18              28.3%

19              10.8%

20              25.2%

21              17.5%

22              27.4%

23              12.9%



I stared at this table for a long time. I thought I had done the simulation

wrong. None of the hands with 17 trumps had a double double fit? After

finding nothing wrong with my code, I tried to think about it

combinatorially.



To have a DDF with 17 total trumps, one of the partnerships has to have an

8-card fit, the other has to have a 9-card fit.



Say east-west have the 8-card fit. Then their second fit also would have to

be an eight-card fit, because we are assuming their 8-card fit was the

longest fit they had. So their combined shape must be 8-8-X-Y where X+Y=10

(X>=Y).



Since NS have a 9-card fit, that means Y must be 4, which in turn means

that X must be 6, which means that NS do not have a double fit.



So it is, in fact, impossible to have a double-double fit when the total

trumps is seventeen.



In fact, looking at the table above, it is significantly less likely that a

DDF will occur whenever the total trumps is an odd number. The graph of the

above data is a zig-zag.



What does this mean, bridge-wise? Well, it means that after the DDF

adjustment, the total tricks value skews towards the odd values.



Here is a table of total tricks, before and after adjustment for DDFs:



       Before Adjustment     After Adjustment

==============================================

14                 10.5%                10.5%

15                 10.8%                10.8%

16                 26.5%                23.2%

17                 22.8%                26.1%

18                 15.5%                11.3%

19                  8.7%                12.0%

20                  3.5%                 3.5%

21                  1.3%                 2.0%

22                  0.3%                 0.5%

==============================================



There is a perversity at this table - the adjustment makes the odd values

more likely than the even values near them. The shape of the graph before

adjustment is pretty straightforward, but after adjustment, it becomes a

more complex beast.



Should this have any affect on your bidding? It's not clear to me, except

that, perhaps, if you think the total tricks is either 16 or 17, you ought

to expect 17, without other information to guide you, but "without other

information..." is a useless bridge phrase - there is always other

information.



Larry Cohen's other adjustments are much harder to determine automatically,

so I haven't been able to throw them into my simulation. What is a rigorous

definition of hand purity, for example?



Last updated June 20, 1995.



Thomas Andrews (thomaso@centerline.com), Copyright 1995.

