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ALGOR.PPA
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1995-08-31
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SUIT DIVISION PROBABILITY ALGORITHM
The probability of suits splitting 3-3, 4-2, 6-0, etc. are well
known. Many books on bridge include a table of such probabilities,
which are helpful in deciding how to play many hands. Faced with
the choice between a finesse and reliance on a 3-3 split, we know
from the probability table that a 3-3 split has only a 35.5%
chance, so we finesse instead--a 50% shot.
There is one thing about these probability tables: They assume you
know nothing significant about the suit distribution in the
opposing hands. Since you usually do have some information about
their hands, maybe only the actual (or probable) distribution of
one or two suits, the probability tables are seldom applicable.
After all, you generally postpone such decisions as finesses or
suit splits as long as possible. By doing so, it is likely that you
learn something about the opposing hands before the critical action
must be taken.
There are other tables not so widely available that provide the
probability of every possible suit split for every given condition
of knowledge concerning the opposing hands. You can find, for
instance, the probability of a 3-3 split in a suit when you know
that another suit is split 6-1 (answer: .28). Few persons possess
such tables, although they are useful for anyone who engages in
postmortem hand analyses.
Lacking those tables, you can work out your own probabilities using
the following algorithm:
1) Make a 3 x 2 matrix, with one horizontal line extended, as
shown:
LHO RHO
┌───────┬───────┐
Known │ │ │
───────┼───────┼───────┼────────
Suit Split│ │ │
├───────┼───────┤
Remainder│ │ │
└───────┴───────┘
LHO and RHO columns are for the left hand opponent's and right hand
opponent's cards, respectively. Enter the number of known sig-
nificant (explained below) cards on the "Known" row. On the "Suit
Split" row enter the suit division for which the probability is to
be calculated. On the "Remainder" row enter the remaining cards
that will make the columns total 13.
Example: You know that LHO has five diamonds, RHO two diamonds, and
that's all you know. You want to know the probability of a 2-4
(LHO-RHO) split in another suit. Adding a 6 and 7 to make each hand
total 13, here is the resultant matrix:
LHO RHO
┌───────┬───────┐
Known │ 5 │ 2 │
───────┼───────┼───────┼────────
Suit Split│ 2 │ 4 │
├───────┼───────┤
Remainder│ 6 │ 7 │
└───────┴───────┘
(2) Working with the 2 x 2 matrix below the long line:
-- Sum the columns and the rows, any order, getting four results:
8, 11, 6, 13
-- Sum the entire 2 x 2 matrix. Here the total is 19.
-- Make a fraction:
Numerator is the product of factorials of the four column/row
sums: 8! x 11! x 6! x 13!
Denominator is the product of the factorials of the four numbers
and the factorial of their total: 2! x 4! x 6! x 7! x 19!
For the example, the resultant fraction is:
8! x 11! x 6! x 13!
─────────────────────── = .34
2! x 4! x 6! x 7! x 19!
The 2-4 split therefore has a 34% probability.
The calculations are easily done on a pocket calculator that has
factorial capability. Many computers include "calculator" software
in their repertoire.
For those who may have forgotten, 0! = 1. For those who have really
forgotten, 4! means 4 x 3 x 2 x 1.
Remember that the known cards must be "significant" cards, which
the ACBL's Encyclopedia calls "positive" cards (as opposed to
"neutral" cards). A significant card is a card that (1) was bound
to have been played, or (2) indicates the position of one or more
other significant cards, or (3) indicates the distribution of all
the outstanding cards of a suit. See the encyclopedia's entry
"CARDS, NEUTRAL AND POSITIVE" for a detailed explanation. You
don't have a copy? You should!