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Bridge Probabilities and Combinatorics

Bridge Probabilities
Combinatorics and Probability Analysis for Bridge Hands

Includes how to calculate the results and computer source code

   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.

Math Symbols/Notation:  Use this link for explanations of the math symbols used. Generally, we will use math notation as expressed/used in Microsoft's Excel spreadsheets.

Bidding Combinatorics:  Statistics  "How to" calculations  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:  Statistics  "How to"  calculations 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  "How to" calculations  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 general:  Statistics  "How to" calculations  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 opponent twice.


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