Bridge probabilities

Bridge probabilities

In the game of bridge mathematical probabilities play a significant role. Different declarer play strategies lead to success depending on the distribution of opponent's cards. To decide which strategy has highest likelihood of success, the declarer needs to have at least an elementary knowledge of probabilities.

The tables below specify the various a priori probabilities, i.e. the probabilities in the absence of any further information. During bidding and play, more information of hands becomes available and requires players to change their probability assumptions.

Probability of suit distributions in two hidden hands

The a priori probabilities that a specific number of cards outstanding in a suit split in the various ways over two hidden hands is given in the table belowH.G. Francis, A.F. Truscott and D.A. Francis (Eds.): The Official Encyclopedia of Bridge, 5th Edition, ISBN 0-943855-48-9.] .


The 39 hand patterns can by classified into four "hand types": balanced hands, three-suiters, two suiters and single suiters. Below table gives the "a priori" likelihoods of being dealt a certain hand-type.

Number of possible deals

In total there are 53,644,737,765,488,792,839,237,440,000 different deals possible, which is equal to 52!/(13!)^4. The immenseness of this number can be understood by answering the question "How large an area would you need to spread all possible bridge deals if each deal would occupy only one square millimeter?". The answer is: "an area more than a hundred million times the total area of the earth".

Obviously, the deals that are identical except for swapping—say—the Hearts2 and the Hearts3 would unlikely give a different result. To make the irrelevance of small cards explicit (which is not always the case though), in bridge such small cards are generally denoted by an 'x'. Thus, the "number of possible deals" in this sense depends of how many non-honour cards (2, 3, .. 9) are considered 'indistinguishable'. For example, if 'x' notation is applied to all cards smaller than ten, then the suit distributions A987-K106-Q54-J32 and A432-K105-Q76-J98 would be considered identical.

The table below [ [ Counting Bridge Deals] , Jeroen Warmerdam] gives the number of deals when various numbers of small cards are considered indistinguishable.

Note that the last entry in the table (37,478,624) corresponds to the number of different distributions of the deck (the number of deals when cards are only distinghuished by their suit).


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