Bridge Card Game

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

$$N={52\choose 13} = 635{,}013{,}559{,}600.$$

where ${n\choose k}$ is a Binomial Coefficient. While the chances of being dealt a hand of 13 Cards (out of 52) of the same suit are

$${4\over {52\choose 13}} = {1\over 158{,}753{,}389{,}900},$$

the chance that one of four players will receive a hand of a single suit is

$${1\over 39{,}688{,}347{,}497}.$$

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 $(1/P)-1:1$.

Hand	Exact Probability	Probability	Odds
13 top honors	${4\over N} = {1\over 158{,}753{,}389{,}900}$	$6.30\times 10^{-12}$	158,753,389,899:1
13-card suit	${4\over N} = {1\over 158{,}753{,}389{,}900}$	$6.30\times 10^{-12}$	158,753,389,899:1
12-card suit, ace high	${4\cdot 12\cdot 36\over N}={4\over 1{,}469{,}938{,}795}$	$2.72\times 10^{-9}$	367,484,697.8:1
Yarborough	${{32\choose 13}\over N}={5{,}394\over 9{,}860{,}459}$	$5.47\times 10^{-4}$	1,827.0:1
four aces	${{48\choose 9}\over N}={11\over 4{,}165}$	$2.64\times 10^{-3}$	377.6:1
nine honors	${{20\choose 9}{32\choose 4}\over N}={888{,}212\over 93{,}384{,}347}$	$9.51\times 10^{-3}$	104.1:1

See also Cards, Poker

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 48-49, 1987.

Kraitchik, M. ``Bridge Hands.'' §6.3 in Mathematical Recreations. New York: W. W. Norton, pp. 119-121, 1942.