Poker

Poker is a Card game played with a normal deck of 52 Cards. Sometimes, additional cards called``jokers'' are also used. In straight or draw poker, each player is normally dealt a hand of five cards. Depending on thevariant, players then discard and redraw Cards, trying to improve their hands. Bets are placed at each discard step. Thenumber of possible distinct five-card hands is

There are special names for specific types of hands. A royal flush is an ace, king, queen, jack, and 10, all of one suit. Astraight flush is five consecutive cards all of the same suit (but not a royal flush), where an ace may count as eitherhigh or low. A full house is three-of-a-kind and a pair. A flush is five cards of the same suit (but not a royal flush orstraight flush). A straight is five consecutive cards (but not a royal flush or straight flush), where an ace may againcount as either high or low.

The probabilities of being dealt five-card poker hands of a given type (before discarding and with no jokers) on theinitial deal are given below (Packel 1981). As usual, for a hand with probability , the Odds against being dealtit are .

Hand	Exact Probability	Probability	Odds
royal flush			649,739.0:1
straight flush			72,192.3:1
four of a kind			4,164.0:1
full house			693.2:1
flush			507.8:1
straight			253.8:1
three of a kind		0.0211	46.3:1
two pair		0.0475	20.0:1
one pair		0.423	1.366:1

Gadbois (1996) gives probabilities for hands if two jokers are included, and points out that it is impossible torank hands in any single way which is consistent with the relative frequency of the hands.

References

Cheung, Y. L. ``Why Poker is Played with Five Cards.'' Math. Gaz. 73, 313-315, 1989.

Conway, J. H. and Guy, R. K. ``Choice Numbers with Repetitions.'' In The Book of Numbers. New York: Springer-Verlag, pp. 70-71, 1996.

Gadbois, S. ``Poker with Wild Cards--A Paradox?'' Math. Mag. 69, 283-285, 1996.

Jacoby, O. Oswald Jacoby on Poker. New York: Doubleday, 1981.

Packel, E. W. The Mathematics of Games and Gambling. Washington, DC: Math. Assoc. Amer., 1981.

• Whitney Singularity
• Whitney Sum
• Whitney Umbrella
• Whittaker Differential Equation
• Whittaker Function
• Whole Number
• Width (Partial Order)
• Width (Size)
• Wiedersehen Manifold
• Wieferich Prime
• Wielandt's Theorem
• Wiener Filter
• Wiener Function
• Wiener-Khintchine Theorem
• Wiener Measure
• Wiener Space
• Wigner 3j-Symbol
• Wigner 6j-Symbol
• Wigner 9j-Symbol
• Wigner-Eckart Theorem
• Wilbraham-Gibbs Constant
• Wilcoxon Rank Sum Test
• Wilcoxon Signed Rank Test
• Wild Knot
• Wild Point