List of poker hands

For nicknames of pocket cards in Texas hold 'em, see List of playing-card nicknames.

In poker, players construct hands of five cards according to predetermined rules, which vary according to which variant of poker is being played. These hands are compared using a hand ranking system that is standard across all variants of poker, the player with the highest-ranking hand winning that particular deal in most variants of poker. In some variants, the lowest-ranking hand can win or tie.

Although used primarily in poker, these hand rankings are also used in some other card games, and in poker dice.

The ranking of a particular hand is increased by including multiple cards of the same card rank, by all five cards being from the same suit, or by all five cards being of consecutive rank. The relative ranking of the various hand categories is based on the probability of being randomly dealt such a hand from a well-shuffled deck.

Contents

General rules

The following rules apply to the ranking of all poker hands. Under typical rules there are 7,462 distinct ranks.

A hand always consists of five cards. In games where more than five cards are available to each player, the best five-card combination of those cards must be played. Any cards not included in the hand do not affect its ranking. For example, if player A holds 3♠ Q♦ and player B holds 3♣ A♣, and five cards 4♣ 5♦ 6♦ 7♠ 10♥ are available to both players, the players hold equally ranking 3-4-5-6-7 straights despite the fact that the player B's ace ranks higher than the player A's queen.

Individual cards are ranked A (highest), K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2 (lowest). Aces can appear low (as if having a value of "1") when part of an A-2-3-4-5 straight or straight flush. In the poker variants ace-to-five and ace-to-six lowball, the ace only plays low, and only plays high in deuce-to-seven lowball. Individual card ranks are used to rank hands that are in the same rank category.

The suits of the cards are used to determine whether a hand forms a flush or straight flush. In most variants, suits do not have an associated value, and play no part in determining the ranking of a hand. Sometimes a ranking called high card by suit is used for randomly selecting a player to deal. Low card by suit usually determines the bring-in better in stud games.

Hands are ranked first by category, then by individual card ranks; even the lowest hand that qualifies in a certain category defeats all hands in all lower categories. For example, 2♦ 2♠ 3♦ 3♣ 4♠, the lowest-valued two pair hand, defeats all hands with just one pair or high card (such as A♠ A♦ K♦ Q♥ J♣). Only between two hands in the same category are card ranks used to break ties.

A poker hand has the same hand ranking regardless of the order in which it is arranged by the deal, by a description, or by a picture. So a hand arranged as 10♠ 8♦ 10♦ 6♣ 10♣ is ranked the same as 10♣ 10♦ 10♠ 8♦ 6♣ even though in the first hand the three of a kind is not immediately obvious.

If there are multiple hands of the same rank at the showdown, the pot is divided equally between the winning players. If the divided number creates fractions (odd chip), the first hand clockwise from the button/dealer gets the chip.

There are 311 875 200 ways (5-permutations) of being dealt five cards from a 52-card deck,[Note 1] but since the order of cards does not matter, there are 5! = 120 5-permutations giving any one hand, so there are only:

{52 \choose 5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = \frac {52 \times 51 \times 50 \times 49 \times 48} { 5!} = 2{,}598{,}960 possible distinct hands.

Hand categories

Straight flush

A straight flush is a hand that contains five cards in sequence, all of the same suit, such as Q♣ J♣ 10♣ 9♣ 8♣(a hand that meets the requirement of both a straight and a flush). Two such hands are compared by their card that is ranked highest. Aces can play low in straights and straight flushes: 5♦ 4♦ 3♦ 2♦ A♦ is a 5-high straight flush, also known as a "steel wheel".[1][2] An ace-high straight flush such as A♦ K♦ Q♦ J♦ 10♦ is known as a royal flush, and is the highest ranking standard poker hand.

There are 40 possible straight flushes, including the four royal flushes. The probability of being dealt one in a five-card deal[Note 2] is \frac {4\cdot 10}{2{,}598{,}960} \approx 0.0015% .

Four of a kind

Four of a kind, also known as quads, is a poker hand such as 9♣ 9♠ 9♦ 9♥ J♥, that contains all four cards of one rank and any other (unmatched) card. Quads with higher ranking cards defeat lower ranking ones. In community-card games (such as Texas Hold 'em) or games with wildcards it is possible for two or more players to obtain the same quad; in this instance, the unmatched card acts as a kicker, so 7♣ 7♠ 7♦ 7♥ J♥ defeats 7♣ 7♠ 7♦ 7♥ 10♣. If two hands have the same kicker, they tie and the pot is split.

There are 624 possible hands including four of a kind; the probability of being dealt one in a five-card deal is \frac {C_{13}^1 \cdot C_{4}^4 \cdot C_{48}^1} {C_{52}^5} = \frac {13 \cdot 1 \cdot 48} {2{,}598{,}960} \approx 0.024%

Full house

A full house, also known as a full boat, is a hand such as 3♣ 3♠ 3♦ 6♣ 6♥, that contains three matching cards of one rank and two matching cards of another rank. Between two full houses, the one with the higher ranking three cards wins, so 7♠ 7♥ 7♦ 4♠ 4♣ defeats 6♠ 6♥ 6♦ A♠ A♣. If two hands have the same three cards (possible in wild card and community card games), the hand with the higher pair wins, so 5♥ 5♦ 5♠ Q♥ Q♣ defeats 5♣ 5♦ 5♠ J♠ J♦. Full houses are described as "Three full of Pair" or occasionally "Three over Pair"; Q♣ Q♦ Q♠ 9♥ 9♣ could be described as "Queens full of nines", "Queens over nines", or simply "Queens full".

There are 3,744 possible full houses; the probability of being dealt one in a five-card hand is \frac {C_{13}^1 C_4^3 \cdot C_{12}^1 C_4^2} {2{,}598{,}960} = \frac{13 \cdot 4 \times 12 \cdot 6}{2{,}598{,}960} \approx 0.14% .

Flush

A flush is a poker hand such as Q♣ 10♣ 7♣ 6♣ 4♣, where all five cards are of the same suit, but not in sequence. Two flushes are compared as if they were high card hands; the highest ranking card of each is compared to determine the winner. If both hands have the same highest card, then the second-highest ranking card is compared, and so on until a difference is found. If the two flushes contain the same five ranks of cards, they are tied and split the pot, that is, suits are not used to rank them.

Flushes are described by their highest card, as in "queen-high flush" to describe Q♦ 9♦ 7♦ 4♦ 3♦. If the rank of the second card is important, it can also be included: K♠ 10♠ 5♠ 3♠ 2♠ is a "king-ten-high flush" or just a "king-ten flush", while K♥ Q♥ 9♥ 5♥ 4♥ is a "king-queen-high flush". In community card games the highest card in the flush may be a community card which is used by multiple players, in which case the flush may be described by the highest non-communal card; in a game with community cards A♣ 10♣ 6♣ 2♣, a player holding Q♣ J♦ would have a "queen-high flush" while a player with K♣ 10♠ holds a "king-high flush"; both players making use of the high ace.

There are 5,148 possible flushes, of which 40 are also straight flushes; the probability of being dealt a flush that is not also a straight flush, in a five-card hand is \frac {4 \cdot C_{13}^5 - 40} {2{,}598{,}960} = \frac {4 \cdot 1{,}287 - 40} {2{,}598{,}960} = \frac {5{,}108} {2{,}598{,}960} \approx 0.20% .

Straight

A straight is a poker hand such as Q♣ J♠ 10♠ 9♥ 8♥, that contains five cards of sequential rank in at least two different suits. Two straights are ranked by comparing the highest card of each. Two straights with the same high card are of equal value, suits are not used to separate them.

Straights are described by their highest card, as in "ten-high straight" or "straight to the ten" for 10♣ 9♦ 8♥ 7♣ 6♠.

A hand such as A♣ K♣ Q♦ J♠ 10♠ is an ace-high straight (also known as Broadway or Royal straight), and ranks above a king-high straight such as K♥ Q♠ J♥ 10♥ 9♣. The ace may also be played as a low card (having a value of "1") in a five-high straight such as 5♠ 4♦ 3♦ 2♠ A♥, which is colloquially known as a wheel. The ace may not "wrap around", or play both high and low; 3♠ 2♦ A♥ K♠ Q♣ is not a straight.

There are 10,240 possible straights, of which 40 are also straight flushes; the probability of being dealt a straight that is not also a straight flush, in a five-card deal is \frac {10 \cdot 4^5 - 40} {2{,}598{,}960} = \frac {10{,}200} {2{,}598{,}960} \approx 0.39% .

Three of a kind

Three of a kind, also called trips or a set, is a poker hand such as 2♦ 2♠ 2♣ K♠ 6♥ that contains three cards of the same rank, plus two cards which are not of this rank nor the same as each other. In Texas hold 'em and other flop games, three of a kind is called a "set" usually when it is composed of a pocket pair and one card of matching rank on the board;[3] It is called "trips" usually when it is made by one card that player has in the hole with two matching cards on the board.[4]

Higher-valued three of a kind defeat lower-valued three of a kind, so Q♠ Q♥ Q♦ 7♠ 4♣ defeats J♠ J♣ J♦ A♦ K♣. If two hands contain three of a kind of the same value, which is possible in games with wild cards or community cards, the kickers are compared to break the tie, so 4♦ 4♣ 4♠ 9♦ 2♣ defeats 4♦ 4♣ 4♠ 8♣ 7♦.

There are 54,912 possible three of a kind hands that are not also full houses or four of a kind; the probability of being dealt one in a five-card deal is \frac {C_{13}^1 C_4^3 \cdot C_{12}^2 C_4^1 C_4^1} {C_{52}^5} = \frac {13 \cdot 4 \cdot 66 \cdot 4 \cdot 4} {2{,}598{,}960} = \frac {54{,}912} {2{,}598{,}960} \approx 2.1% .

Two pair

A poker hand such as J♥ J♣ 4♣ 4♠ 9♥, that contains two cards of the same rank, plus two cards of another rank (that match each other but not the first pair), plus any card not of either rank, is called two pair. To rank two hands both containing two pair, the higher ranking pair of each is first compared, and the higher pair wins (so 10♠ 10♣ 8♥ 8♣ 4♠ defeats 8♥ 8♣ 4♠ 4♣ 10♠). If both hands have the same top pair, then the second pair of each is compared, such that 10♠ 10♣ 8♥ 8♣ 4♠ defeats 10♠ 10♣ 4♠ 4♥ 8♥. If both hands have the same two pairs, the kicker determines the winner, so 10♠ 10♣ 8♥ 8♣ A♦ beats 10♠ 10♣ 8♥ 8♣ 4♠.

Two pair are described by the higher pair first, followed by the lower pair if necessary; K♣ K♦ 9♠ 9♥ 5♥ is described as "Kings over nines", "Kings and nines", or simply "Kings up" if the nines are not important.

There are 123,552 possible two pair hands that are not also full houses; the probability of being dealt one in a five-card deal is \frac {C_{13}^2 \cdot C_4^2 C_4^2 \cdot C_{44}^1} {C_{52}^5} = \frac {78 \cdot 6 \cdot 6 \cdot 44} {2{,}598{,}960} = \frac {123{,}552} {2{,}598{,}960} \approx 4.75% .

One pair

One pair is a poker hand such as 4♥ 4♠ K♠ 10♦ 5♠, that contains two cards of one rank, plus three cards which are not of this rank nor the same as each other. Higher ranking pairs defeat lower ranking pairs; if two hands have the same pair, the non-paired cards (the kickers) are compared in descending order to determine the winner.

There are 1,098,240 possible one pair hands; the probability of being dealt one in a five-card deal is \frac {C_{13}^1 C_4^2 \cdot (C_{12}^3 \cdot 4^3)} {C_{52}^5} = \frac {13 \cdot 6 \cdot (220 \cdot 64)} {2{,}598{,}960} = \frac {1{,}098{,}240} {2{,}598{,}960} \approx 42.25% .

High card

A high-card or no-pair hand is a poker hand such as K♥ J♥ 8♣ 7♦ 4♠, made of any five cards not meeting any of the above requirements. Essentially, no hand is made, and the only thing of any potential meaning in the player's hand is their highest card. Nevertheless, they sometimes win a pot if the other players fold or even at a showdown. Two high-card hands are ranked by comparing the highest ranking card. If those are equal, then the next highest ranking card from each hand is compared, and so on until a difference is found.

High card hands are described by the one or two highest cards in the hand, such as "king high", "ace-queen high", or by as many cards as are necessary to break a tie. They are also referred to as "nothing", "garbage", and other derogatory terms.

The lowest possible high card is seven-high (such as 7♠ 5♣ 4♦ 3♦ 2♣), because a hand such as 6♦ 5♣ 4♠ 3♦ 2♥ would be a straight, and in 6♦ 5♣ 4♠ 3♦ A♥ the ace would serve as the high card.

Of the 2,598,960 possible hands, 1,302,540 do not contain any pairs and are neither straights nor flushes. As such, the probability of being dealt "no pair" in a five-card deal is \frac {1{,}302{,}540} {2{,}598{,}960} \approx 50% .

Variations

Decks using a wild card or bug

The use of a joker as a bug creates a variation to game play. When a joker is introduced, it most commonly functions as a fifth ace, unless it can be used to complete a flush or straight. Some casino draw poker variants use a joker, and thus the best possible hand is five of a kind aces, or A♥ A♦ A♣ A♠ J with the joker representing the fifth ace. In lowball, the joker plays as the lowest card not already in the hand. In non-casino settings the joker is sometimes a full wild card.

The odds of being dealt a five of a kind (of any kind) in a five-card deal is ~0.0005%, three times less likely than a straight flush. The odds of being dealt five of a kind aces is ~0.0000348%—one in every 2,869,685 deals.

Lowball

Main article: Lowball (poker)

Some variants of poker called lowball or low poker are played with the standard hand rankings described here, but with the objective reversed: players strive not for the highest ranking of the above hands but for the lowest ranking hand. There are several methods of ranking low hands, including ace-to-five low, deuce-to-seven low, ace-to-six low and deuce-to-six low. The ace-to-five (known as California lowball) is used in razz, in some draw poker varieties, and in deuce-to-seven (known as Kansas City lowball).

Stripped deck variant

Five-card draw poker is sometimes played with a stripped deck. This variant is commonly known as "seven-to-ace" or "ace-to-seven" (abbreviated as A-7 or 7-A). It can be played by up to five players. When four or fewer players play, a normal 32-card deck without jokers, with ranks ranging from ace to seven, is used. With five players, the sixes are added to make a 36-card deck. The deck thus contains only eight or nine different card ranks, compared to 13 in a standard deck. This affects the probabilities of making specific hands, so a flush ranks above a full house and below four of a kind in the seven-to-ace five-card draw.

See also

Notes

  1. ^ In general, the number of k-permutations taken from a set of n is
    n^{\underline k}=n\times(n-1)\times\cdots\times(n-k%2B1),
    which in this case gives 52\times51\times50\times49\times48 = 311\,875\,200
  2. ^ Note that the probabilities will vary slightly in games where the player can select from more than five cards (such as Texas Hold'em and Omaha), as the player is able to select the best possible hand from the cards available.

References

  1. ^ "Glossary". PokerTips.com. http://www.pokertips.org/glossary/w/STEEL+WHEEL. Retrieved 2007-12-20. 
  2. ^ "Poker dictionary". PokerWorks.com. Archived from the original on 2007-12-27. http://web.archive.org/web/20071227073226/http://pokerworks.com/poker-dictionary/steel-wheel.html. Retrieved 2007-12-20. 
  3. ^ Sklansky, David. Small Stakes Hold 'Em (1 ed.). p. 127. 
  4. ^ "Trips Poker Term -- Defintion of the Poker Term Trips". About.com. http://poker.about.com/od/pokerglossary/g/Trips.htm. Retrieved 13 September 2011. 
  • Elkan Allan & Hannah Mackay (2007-10-18). The Poker Encyclopedia. Anova. ISBN 1906032092. 
  • Krieger, Lou (2006). The Poker Player's Bible. Struik. p. 249. ISBN 9781770074699. 
  • Wolpin, Stewart (1990). The Rules of Neighborhood Poker According to Hoyle. New Chapter Press. p. 335. ISBN 9780942257199. 

External links