Poker Hand Categories

Straight flush

"Straight flush" redirects here. For the World War II bomber, see Straight Flush (B-29). For NATO designation of the Soviet radar, see 2K12 Kub.

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 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".^[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.

In five-card poker, there are 40 possible straight flushes, including the 4 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]}

[edit]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 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]}

[edit]Full house

"Full house" redirects here. For other uses, see Full house (disambiguation).

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 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".

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 5 cards out of the 52 cards in the deck. Numerator: first we multiply the number of ways to choose 1 rank out of the 13 ranks, then 3 cards out of 4 of that chosen rank. Second, we choose 1 out of 12 reminding ranks, and choose 2 cards out of 4 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]}

[edit]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]}

[edit]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". The ace may not "wrap around", or play both high and low; 3♠ 2♦ A♥ K♠ Q♣ is not a straight.

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]}

[edit]Three of a kind

"Three of a kind" redirects here. For other uses, see Three of a Kind (disambiguation).

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;^[5] 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.^[6]

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, so4♦ 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]}

[edit]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 over nines", "Kings and 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]}

[edit]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 that in 5-card poker, one pair hands are more frequent than "no pair" hands in 7-card poker.^[4]^{[Note 3]}

[edit]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 in6♦ 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%.

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