Bridge is a card game played with a normal deck of 52 cards. The number of possible distinct 13-card hands is

 N=(52; 13)=635013559600,

where (n; k) is a binomial coefficient. While the chances of being dealt a hand of 13 cards (out of 52) of the same suit are

 4/((52; 13))=1/(158753389900)

(Mosteller 1987, p. 8), the chance that at least one of four players will receive a hand of a single suit is

 (18772910672458601)/(745065802298455456100520000) approx 2.519631234×10^(-11),

while the probability that exactly one person will have a perfect hand is

 (242753155112819)/(9634471581445544690955000) approx 2.519631233×10^(-11)

(Gridgeman 1964; Mosteller 1987, p. 8).

There are special names for specific types of hands. A ten, jack, queen, king, or ace is called an "honor." Getting the three top cards (ace, king, and queen) of three suits and the ace, king, and queen, and jack of the remaining suit is called 13 top honors. Getting all cards of the same suit is called a 13-card suit. Getting 12 cards of same suit with ace high and the 13th card not an ace is called 2-card suit, ace high. Getting no honors is called a Yarborough.

The probabilities of being dealt 13-card bridge hands of a given type are given below. As usual, for a hand with probability P, the odds against being dealt it are (1/P)-1:1.

handexact probabilityprobabilityodds
13 top honors4/N=1/(158753389900)6.30×10^(-12)158753389899:1
13-card suit4/N=1/(158753389900)6.30×10^(-12)158753389899:1
12-card suit, ace high(4·12·36)/N=4/(1469938795)2.72×10^(-9)367484697.8:1
Yarborough((32; 13))/N=(5394)/(9860459)5.47×10^(-4)1827.0:1
four aces((48; 9))/N=(11)/(4165)2.64×10^(-3)377.6:1
nine honors((20; 9)(32; 4))/N=(888212)/(93384347)9.51×10^(-3)104.1:1

See also

Bridged Graph, Bridgeless Graph, Cards, Graph Bridge, Poker

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 48-49, 1987.Bizley, M. T. L. Letter to the Editor. Amer. Stat. 18, 31, Apr. 1964.Gridgeman, N. T. "The Mystery of the Missing Deal." Amer. Stat. 18, 15-16, Feb. 1964.Kraitchik, M. "Bridge Hands." §6.3 in Mathematical Recreations. New York: W. W. Norton, pp. 119-121, 1942.Mosteller, F. "Perfect Bridge Hand." Problem 8 in Fifty Challenging Problems in Probability with Solutions. New York: Dover, pp. 2 and 22-24, 1987.Norton, H. W. Letter to the Editor. Amer. Stat. 18, 30-31, Apr. 1964.Reese, T. Bridge for Bright Beginners. New York: Dover, 1973.Rubens, J. The Secrets of Winning Bridge. New York: Dover, 1981.

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Cite this as:

Weisstein, Eric W. "Bridge." From MathWorld--A Wolfram Web Resource.

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