List of poker hands

Examples of poker hand categories in descending order

In poker, players construct hands of playing 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 wins that particular deal in most poker games. In some variants, the lowest-ranking hand can win or tie.

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 the five cards forming a consecutive series. 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.

General rules

The following rules apply to the ranking of all poker hands unless specifically changed by game variant rules or house rules.

A poker hand consists of exactly five cards; no more, no fewer. 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 bettor 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 necessarily immediately obvious.

If there are multiple hands of the same rank at the showdown (therefore no kicker(s), or kickers have equal ranks), the pot is divided equally between the winning players. If the divided number creates fractions (odd chip), the odd chip is awarded by house rules. Typically, in draw and community card games, this award is to the winning first hand clockwise from the button/dealer; in stud games it is to the player with highest card by suit in hand.

There are 311,875,200 ways (5-permutations)^{[Note 1]} of being dealt five cards from a 52-card deck, but since the order of cards does not matter, there are 5! = 120 5-permutations given in any one hand, so there are only 2,598,960 ( $\begin{matrix} {52 \choose 5} \end{matrix}$ ) possible combinations in five-card poker. Similarly, in seven-card poker there are 133,784,560 ( $\begin{matrix} {52 \choose 7} \end{matrix}$ ) combinations.^{[Note 2]}

If the game is played without the high card by suit rule, hands that have equivalent values are considered to be the same hand, there will be 7,462 distinct hands in five-card poker, and 4,824 distinct hands when choosing the best five in seven-card poker, like Texas hold 'em; the number of seven-card poker hands is fewer than of five-card poker because some hands are impossible with seven cards (e.g. 7-high).^[1]

Hand categories

Straight flush

Defeats

Ties with

Straight flush examples

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 requirements of both a straight and a flush). Two such hands are compared by their card that is ranked highest. Aces can play high or low in straights and straight flushes: 5♦ 4♦ 3♦ 2♦ A♦ is a 5-high straight flush, also known as a "steel wheel".^[2]^[3]

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. It is usually treated as a distinct hand in video poker^{[citation needed]}. There are four possible royal flushes, one of each suit.

In five-card poker, there are 40 possible straight flushes, including the four royal flushes. The probability of being dealt a straight flush is $\frac {4\cdot 10}{2{,}598{,}960} \approx 0.0015\%$ .

In seven-card poker such as Texas hold 'em, the frequency of straight flush is 41,584 (4,324 for royal flush and 37,260 for non-royal straight flush); the probability of straight flush is approximately 0.0311% (0.0032% for royal flush and 0.0279% for non-royal straight flush).^[4]^{[Note 3]}

Four of a kind

Defeats

Four of a kind examples

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 or multiple decks 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. In some countries the term Carré is used.

In five-card poker, there are 624 possible hands including four of a kind, the probability of being dealt one is $\frac {C_{13}^1 C_{4}^4 \cdot C_{12}^1 C_{4}^1} {C_{52}^5} = \frac {13 \cdot 1 \cdot 12 \cdot 4} {2{,}598{,}960} \approx 0.024\%$ .

In seven-card poker, the frequency of four of a kind is 224,848, the probability of being dealt one is approximately 0.168%.^[4]^{[Note 3]}

Full house

Defeats

Full house examples

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 in their hand 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 "full house, threes over pairs"; thus, Q♣ Q♦ Q♠ 9♥ 9♣ could be described as "queens full of nines" (or simply "queens full" if no mention of the pair as tiebreaker is necessary).

In five-card poker, there are 3,744 possible full houses; the probability of being dealt one is $\frac {C_{13}^1 C_4^3 \cdot C_{12}^1 C_4^2} {C_{52}^5} = \frac{13 \cdot 4 \times 12 \cdot 6}{2{,}598{,}960} \approx 0.1441 \%$ .

An explanation of the formula: there are 13 ranks (ace through king), and four of each rank (one per suit), in the standard 52-card deck. Denominator: we express choosing five cards out of the 52 cards in the deck. Numerator: first we multiply the number of ways to choose one rank out of the 13, then three cards out of four of that chosen rank. Second, we choose one out of 12 remaining ranks, and choose two cards out of four from that rank.

In seven-card poker, the frequency of full house is 3,473,184, the probability of being dealt one is approximately 2.60%.^[4]^{[Note 3]}

Flush

Defeats

Flush examples

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.

In five-card poker, 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 is $\frac {4 \cdot C_{13}^5 - 40} {C_{52}^5} = \frac {4 \cdot 1{,}287 - 40} {2{,}598{,}960} = \frac {5{,}108} {2{,}598{,}960} \approx 0.196\%$ .

In seven-card poker, the frequency of flush is 4,047,644, the probability of being dealt one is approximately 3.03%.^[4]^{[Note 3]}

Straight

Defeats

Ties with

Straight examples

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" or Baby Straight. The ace may not "wrap around", or play both high and low; 3♠ 2♦ A♥ K♠ Q♣ is not a straight. Drawing to an "Inside Straight" is the case wherein the player does not hold 4 sequential straight cards which can be completed by a card at either 'end', but rather 4 cards with a gap so that only 1 card can complete the straight. For example, 4-5-6-8 can only be competed with an 'inside' 7 (also called 'gut-shot'), whereas 4-5-6-7 can be completed by either a 3 or an 8 (also called 'open-ended')...^[5]

In five-card poker, 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 is $\frac {10 \cdot 4^5 - 40} {2{,}598{,}960} = \frac {10{,}200} {2{,}598{,}960} \approx 0.39\%$ .

In seven-card poker, the frequency of straight is 6,180,020, the probability of being dealt one is approximately 4.62%.^[4]^{[Note 3]}

Three of a kind

Defeats

Three of a kind examples

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;^[6] 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.^[7]

A higher-valued three-of-a-kind defeats a lower-valued three-of-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♦.

In five-card poker, 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 is $\frac {C_{13}^1 C_4^3 \cdot C_{12}^2 C_4^1 C_4^1} {C_{52}^5} = \frac {13 \cdot 4 \cdot 66 \cdot 4 \cdot 4} {2{,}598{,}960} = \frac {54{,}912} {2{,}598{,}960} \approx 2.11\%$ .

In seven-card poker, the frequency of three of a kind is 6,461,620, the probability of being dealt one is approximately 4.83%.^[4]^{[Note 3]}

Two pair

Defeats

Two pairs examples

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 and nines", "kings over nines", or simply "Kings up" if the nines are not important.

In five-card poker, there are 123,552 possible two pair hands that are not also three of a kind hands or higher hands; the probability of being dealt one is $\frac {C_{13}^2 C_4^2 C_4^2 \cdot C_{11}^1 C_{4}^1} {C_{52}^5} = \frac {78 \cdot 6 \cdot 6 \cdot 11 \cdot 4} {2{,}598{,}960} = \frac {123{,}552} {2{,}598{,}960} \approx 4.75\%$ .

In seven-card poker, the frequency of two pair is 31,433,400, the probability of being dealt one is approximately 23.5%.^[4]^{[Note 3]}

One pair

Defeats

One pair examples

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.

In five-card poker, there are 1,098,240 possible one pair hands; the probability of being dealt one is $\frac {C_{13}^1 C_4^2 \cdot (C_{12}^3 \cdot 4^3)} {C_{52}^5} = \frac {13 \cdot 6 \cdot (220 \cdot 64)} {2{,}598{,}960} = \frac {1{,}098{,}240} {2{,}598{,}960} \approx 42.26\%$ .

In seven-card poker, the frequency of one pair is 58,627,800, the probability of being dealt one is approximately 43.8%. Unlike in 5-card poker, in 7-card poker one-pair hands are more frequent than "no-pair" hands.^[4]^{[Note 3]}

High card

Defeats

High card examples

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 hand is the 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♦ 4♠ 3♦ 2♣ A♥ the ace would serve as the high card.

Of the 2,598,960 possible five-card combinations, $(C^5_{13} - 10)(4^5 - 4)=1{,}302{,}540$ 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 $\frac {1{,}302{,}540} {2{,}598{,}960} \approx 50.11\%$ .

In seven-card poker, the frequency of such "no pair" is 23,294,460; the probability of being dealt one is approximately 17.4%.^[4]^{[Note 3]}

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♠ Jkr 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.^{[clarification needed]}

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

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).

Notes

↑ In general, the number of k-permutations taken from a set of n is

$n^{\underline k} = \frac {n!}{(n-k)!}$ ,

which in this case gives $\frac {52!}{(52-5)!} = \frac {52!}{47!} = 52\times51\times50\times49\times48 = 311,875,200$
↑ ${52 \choose 5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = \frac {52 \times 51 \times 50 \times 49 \times 48} { 5 \times 4 \times 3 \times 2 \times 1} = 2{,}598{,}960$ ;
${52 \choose 7} = \frac{52!}{7!(52-7)!} = \frac{52!}{7!45!} = \frac {52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46} { 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 133{,}784{,}560$
↑ ^3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 ^3.6 ^3.7 ^3.8 See also Poker probability#Derivation of frequencies of 7-card poker hands for the derivation. This can apply to Texas hold 'em and seven-card stud. Note that the probabilities vary in Omaha; the derivation of frequencies of Omaha hands is different from 7-card or 9-card poker hands.

References

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External links

Printable chart of poker hand rankings (.pdf format)

[1] In general, the number of k-permutations taken from a set of n is

$n^{\underline k} = \frac {n!}{(n-k)!}$ ,

which in this case gives $\frac {52!}{(52-5)!} = \frac {52!}{47!} = 52\times51\times50\times49\times48 = 311,875,200$

[2] ${52 \choose 5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = \frac {52 \times 51 \times 50 \times 49 \times 48} { 5 \times 4 \times 3 \times 2 \times 1} = 2{,}598{,}960$ ;
${52 \choose 7} = \frac{52!}{7!(52-7)!} = \frac{52!}{7!45!} = \frac {52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46} { 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 133{,}784{,}560$

[7cardprob-7] 3.0 ^3.1 ^3.2 ^3.3 ^3.4 ^3.5 ^3.6 ^3.7 ^3.8 See also Poker probability#Derivation of frequencies of 7-card poker hands for the derivation. This can apply to Texas hold 'em and seven-card stud. Note that the probabilities vary in Omaha; the derivation of frequencies of Omaha hands is different from 7-card or 9-card poker hands.

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v t e Poker
Index of poker articles
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Category Commons WikiProject

List of poker hands

Contents

General rules

Hand categories

Straight flush

Four of a kind

Full house

Flush

Straight

Three of a kind

Two pair

One pair

High card

Variations

Decks using a wild card or bug

Lowball

See also

Notes

References

External links

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