The following sections cover several aspects of
Bridge probabilities and combinatorics. Each section has a
link that gives the statistical results and another link that
shows how the results are calculated. The "How to" sections
give both a generalized description of the calculations and
algorithm as well as the "C" source code.
Use this link for
explanations of the math symbols used. Generally, we will use
math notation as expressed/used in Microsoft's Excel
Bidding Combinatorics: Statistics
There are 1.28746 E+47 (Scientific
notation for 128+ billion billion billion billion billion
(American billion = 1,000,000,000)) different ways to bid
after the cards have been dealt. Most of these sequences are
nonsensical, but they are legal, hence they must be counted.
This is about 2.4 billion billion times larger than the number
of ways that four hands can be dealt from a deck of cards.
(Total number of possible deals = FACT(52) / ((FACT(13)^4) =
5.36447 E+28) (Note: The order of the cards in a bridge hand
is not relevant.)
Bidding combinatorics for a hand is divided into
3 parts. Part one is just 0 to 3 "Passes" before someone
mentions a quantity (1 - 7) and a suit (or No Trump). Part 2
contains all quantity and suit bids (We will count suit bids
for the stats output) through the last "quantity-suit" bid.
This will include all possible intervening bids of "pass",
"double", and "redouble". Part 3 comes after the last
"quantity-suit" bid, and the only words allowed are "pass",
"double", and "redouble".
Suit Distribution Combinatorics:
It would be nice to be dealt all
13 cards of a suit, but it hasn't happened to me yet. (In
practice, if I bid 7 "whatevers", the next person would steal
the hand with a bid of 7 No Trump. Worse yet they would know
the distribution and just might make it.) The most
common suit distribution is 4, 4, 3, 2 and the mundane 4, 3,
3, 3 is actually only the 5th most common.
Point Count Combinatorics: Statistics
A guide to the strength of any
given hand is frequently determined by the "Old reliable"
point count. Using this analysis, a player counts 4 points for
every Ace in his hand, 3 points for Kings, 2 points for
Queens, and 1 point for Jacks. The total becomes the point
count. A hand that has no honors (not even a 10) is frequently
referred to as a "Yarborough" (9 high or worse). There are
347,373,600 "Yarboroughs" (COMBIN(32, 13)), and I think I'm
already working on the second half.
"With 8 ever, with 9 never" - Splits in
This old rule of thumb is used to
determine whether you should take a finesse or play for the
drop given that you have 8 or 9 cards in a suit (your hand
plus the dummy), but are missing the Queen. The statistical
difference is quite small for the "With 9 never" portion, and
is usually far outweighed by knowledge gained from bidding,
card play, or just plain keeping the dangerous opponent from
leading through your broken strength in other suits.
Nevertheless we will give the numbers, and you as a player can
see things in perspective.
Solutions to the round-robin Bridge tournament
problem (Whist Tournament Schedules)
Cyclic solutions for (4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24,
25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53,
56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84,
85, 88, 92, 96 players) up to 24 Bridge tables such that each
player has each other player as a partner once, and as an
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