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15-17 NT
Flannery
20-21 NT
Pass Flat Hand

I am going to write a book on computer analysis of bridge hands. Here is the abstract of the introduction

Introduction 

Random event factor for games 

There are games which are completely determined by the skill of the players. Chess is a great example of this type of games. In theory if a player or a computer program which can play chess a lot better then you, then they will always win against you. On the other hand there are games which are completely determined by random events. Keno, lottery and flipping of a coin are examples of this type of game. In theory the best player or computer program cannot do any better then the worse player in this type of game. Bridge is somewhere in between. It is one of the most complicated games for a computer program to play because there are many cards cannot be seen by the players. Different phases of the game of bridge have different randomness factors. There is more randomness factor in the bidding phase due to only 13 cards are known to each player.  During the play of the hand, once the opening lead is done players can see more cards because the dummy’s hand is exposed to all the players. The amount of randomness is thus reduced. In some cases a single dummy program can find a line of play that will always make a contract or the maximum number of tricks. But for most hands the randomness still is significant and the outcome of the hand depends on how the suits split and where the key honor cards is located.   

Double dummy analysis of bridge hands 

This is the technique in solving how many tricks can be make for NT S H D C contracts when all 52 cards are exposed to all players. It assumes each player will make the best play possible when they select their cards to play. A double dummy database consists of 20 numbers for each bridge hand. The number of tricks the 4 declarers (north, south, east and west) can make in the 5 different types of contracts (spade, heart, diamond, club and no trump). From these 20 numbers the computer program can determine what the best score for the hand is, can a game or slam be made. The best score from the double dummy analysis is call par for the hand like in the game of gulf. This par value may not be a good number to use if you only look at one or two hands. If the only thing that is in question is which way to finesse a queen, the double dummy result will always find the correct direction. But on the other hand there are many hands the opening lead will give the declarer an extra trick which the double dummy result will never make that “mistake”.  If you use a large enough sample of bridge hands the double dummy par value for the hands are quite accurate and very realistic to be used in deciding what one should do. After studying hundreds of computer simulation results and thousands of hands manually, I notice the following small adjustment. The par value of double dummy analysis is a little bit higher then single dummy analysis for slam hands.  This is due to more chances for two way finesse of queens and jacks and the dropping of singleton k and doubleton queens, and opening leads which do not give away a trick against slams is also a little bit easier to find.  

How random is the game of bridge 

We were told when we first start to play bridge that it takes about 26 points to make a game in NT or in the majors, about 29 points to make a minor suit game (11 tricks), about 33 points to make a small slam, and about 37 points to make a grand slam.  Out of  a random 1.4 million bridge hands, about 4.6% of the hands have 29 points between north and south. About 35% of the hands make 11 tricks.  38% make 10 tricks, 14% make 9 tricks and 9% make 12 tricks. Out of the same 1.4 million hands, about 3.8% of the hands have 30 points between north and south. About 43% of the hands make 11 tricks.  31% make 10 tricks, 8% make 9 tricks and 16% make 12 tricks. The above numbers show that bridge has a very high degree of randomness. On top of the above randomness, you can never be sure how many points you have between you and your partner. In most cases you will be off by one to three points.  But even if you know exactly how many points your partnership has, you still have very diverse results.  

Computer analysis give you a statistical advantage 

Computer analysis of bridge hands can confirm or dismiss many common statements in the game of bridge. There are about 250,000 people claim they are at the NBA basketball game when Wilt Chamberlain scored 100 points in one game. Many bridge players insist they have fewer points then their opponents most of the time. “Every time when I raise my partner’s one club bid, my partner has 3 clubs.” The first two is obvious not true.  The last one may or may not be true. If the player always thinks his partner has 3 clubs when open with 1 club, and never raises in club unless he has 6 clubs, then the statement is very close to reality.  On the other hand if he raises in club with 4 or more clubs then the statement is not true. Computer analysis will give you solid statistical results which you can apply to your bridge game. Computer analyses which do not involve double dummy analysis are 100% accurate. I just run a computer simulation of 1 million hands; north open 1 club and south have 4 clubs. North has 3 clubs 25%, 4 clubs 34%, 5 clubs 29% 6 or more clubs 11% with an average of 4.26.   For any practical purpose of playing the game of bridge this information is 100% accurate. I also search the double dummy database for north opening 1c and south have exactly 6 points and 4 clubs. The average number of tricks taken by the north south partnership is 8.3 tricks. So if the bidding go 1c / 1s /?.  South should bid 2c and not worry about partner having only 3 clubs. But the trick taking value is not 100% accurate. If the database is a single dummy database, then the result is 100% accurate. I am not aware of any mathematical or theoretical proof of how well double dummy data correlate to single dummy data. General perception is that double dummy data are very close to single dummy data with a large sample of hands. It is very difficult to compute good single dummy data for a large number of bridge hands.  What we can do today is use double dummy database.