# List of poker hands

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In poker, players construct hands of five cards according to predetermined rules, which vary according to the precise variant of poker being played. These hands are compared using a standard ranking system, and the player with the highest-ranking hand wins that particular deal. Although used primarily in poker, these hand rankings are also used in other card games, and with poker dice.

The strength of a hand is increased by having multiple cards of the same rank, all the cards being from the same suit, or having all the cards with consecutive values. The position of the various possible hands is based on the probability of being randomly dealt such a hand from a well-shuffled deck.

## General rulesEdit

The following general rules apply to evaluating poker hands, whatever set of hand values are used.

• Individual cards are ranked A (high), K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2, A. Aces only appear low when part of an A-2-3-4-5 straight or straight flush. Individual card ranks are used to compare hands that contain no pairs or other special combinations, or to compare the kickers of otherwise equal hands. The ace plays low only in ace-to-five and ace-to-six lowball games, and plays high only in deuce-to-seven lowball.
• Suits have no value. The suits of the cards are mainly used in determining whether a hand fits a certain category (specifically the flush and straight flush hands). In most variants, if two players have hands that are identical except for suit, then they are tied and split the pot (so Template:Cards does not beat Template:Cards). 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.
• 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 plays.
• Hands are ranked first by category, then by individual card ranks: even the lowest qualifying hand in a certain category defeats all hands in all lower categories. The smallest two pair hand (Template:Cards), for example, defeats all hands with just one pair or high card. Only between two hands in the same category are card ranks used to break ties.

## Standard rankingEdit

For ease of recognition, poker hands are usually presented with the most important cards on the left, with cards descending in importance towards the right. However, a poker hand still has the same value however it is arranged. There are 311,875,200 ways ("permutations") of being dealt five cards from a 52 card deck,[1] but since the order of cards does not matter there are ${52 \choose 5} = \frac {311,875,200}{5!} = \frac {311,875,200}{120} = 2,598,960$ possible distinct hands ("combinations").

### Straight flushEdit

A straight flush is a poker hand which contains five cards in sequence, all of the same suit, such as Template:Cards. Two such hands are compared by their highest card; since suits have no relative value, two otherwise identical straight flushes tie (so Template:Cards ties with Template:Cards). Aces can play low in straights and straight flushes: Template:Cards is a 5-high straight flush, also known as a "steel wheel".[2][3] An ace-high straight flush such as Template:Cards 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 a straight flush is $\frac {40} {52 \times 51 \times 50 \times 49 \times 48 \div 5!} = \frac {40}{2,598,960} \approx 0.00154%$

### Four of a kindEdit

Template:Imageframe Four of a kind, also known as quads, is a poker hand such as Template:Cards, which contains four cards of one rank, and an unmatched card of another rank. It ranks above a full house and below a straight flush. Higher ranking quads 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 Template:Cards defeats Template:Cards.

There are 624 possible hands including four of a kind; the probability of being dealt one is $\frac {624} {2,598,960} \approx 0.024%$

### Full houseEdit

Template:Imageframe A full house, also known as a full boat, is a hand such as Template:Cards, which contains three matching cards of one rank, and two matching cards of another rank. It ranks below a four of a kind and above a flush. Between two full houses, the one with the higher ranking set of three wins, so Template:Cards defeats Template:Cards. If two hands have the same set of three (possible in wild card and community card games), the hand with the higher pair wins, so Template:Cards defeats Template:Cards. Full houses are described as "Three full of Pair" or occasionally "Three over Pair"; Template:Cards could be described as "Queens over nines", "Queens full of nines", or simply "Queens full". However, "Queens over nines" is more commonly used to describe the hand containing two pairs, one pair of queens and one pair of nines, as in Template:Cards.

There are 3,744 possible full houses; the probability of being dealt one in a five-card hand is $\frac {3744} {2,598,960} \approx 0.14%$

### FlushEdit

Template:Imageframe A flush is a poker hand such as Template:Cards, which contains five cards of the same suit, not in rank sequence. It ranks above a straight and below a full house. 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 – suits are not used to differentiate them. Flushes are described by their highest card, as in "queen-high flush" to describe Template:Cards. If the rank of the second card is important, it can also be included: Template:Cards is a "king-ten-high flush" or just a "king-ten flush", while Template:Cards is a "king-queen-high flush".

There are 5,148 possible flushes, of which 40 are also straight flushes; the probability of being dealt a flush in a five-card hand is $\frac {5108} {2,598,960} \approx 0.20%$

### StraightEdit

Template:Imageframe A straight is a poker hand such as Template:Cards, which contains five cards of sequential rank but in more than one suit. It ranks above three of a kind and below a flush. 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 Template:Cards.

A hand such as Template:Cards is an ace-high straight (also known as Broadway), and ranks above a king-high straight such as Template:Cards. The ace may also be played as a low card in a five-high straight such as Template:Cards, which is colloquially known as a wheel. The ace may not "wrap around", or play both high and low: Template:Cards is not a straight, but just ace-high no pair.

There are 10,240 possible straights, of which 40 are also straight flushes; the probability of being dealt a straight in a five-card hand is $\frac {10,200} {2,598,960} \approx 0.39%$

### Three of a kindEdit

Template:Imageframe Three of a kind, also called trips, set or a prile (the last of these from its use in three card poker[4]), is a poker hand such as Template:Cards, which contains three cards of the same rank, plus two unmatched cards. It ranks above two pair and below a straight. In Texas hold 'em and other flop games, a "set" refers specifically to a three of a kind composed of a pocket pair and one card of matching rank on the board (as opposed to two matching cards on the board and a third in the player's hand).[5] Higher-valued three of a kind defeat lower-valued three of a kind, so Template:Cards defeats Template:Cards. If two hands contain threes of a kind of the same value, possible in games with wild cards or community cards, the kickers are compared to break the tie, so Template:Cards defeats Template:Cards.

There are 54,912 possible three of a kind hands which are not also full houses; the probability of being dealt one in a five-card hand is $\frac {54,912} {2,598,960} \approx 2.11%$

### Two pairEdit

Template:Imageframe A poker hand such as Template:Cards, which contains two cards of the same rank, plus two cards of another rank (that match each other but not the first pair), plus one unmatched card, is called two pair. It ranks above one pair and below three of a kind. To rank two hands both containing two pair, the higher ranking pair of each is first compared, and the higher pair wins (so Template:Cards defeats Template:Cards). If both hands have the same "top pair", then the second pair of each is compared, such that Template:Cards defeats Template:Cards). Finally, if both hands have the same two pairs, the kicker determines the winner: Template:Cards loses to Template:Cards. Two pair are described by the higher pair first, followed by the lower pair if necessary; Template:Cards could be 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 hand is $\frac {123,552} {2,598,960} \approx 4.75%$

### One pairEdit

Template:Imageframe One pair is a poker hand such as Template:Cards, which contains two cards of the same rank, plus three other unmatched cards. It ranks above any high card hand, but below all other poker hands. 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 hand is $\frac {1,098,240} {2,598,960} \approx 42.27%$

### High cardEdit

Template:Imageframe A high-card or no-pair hand is a poker hand such as Template:Cards, in which no two cards have the same rank, the five cards are not in sequence, and the five cards are not all the same suit. It is also referred to as "no pair", as well as "nothing", "garbage," and various other derogatory terms. High card ranks below all other poker hands; two such 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.

The lowest possible high card is seven-high (such as Template:Cards), because a hand such as Template:Cards would be a straight.

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 hand is $\frac {1,302,540} {2,598,960} \approx 50.17%$

## VariationsEdit

### Decks using a bugEdit

The use of a joker as a bug creates a slight 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. Normally casino draw poker variants use a joker, and thus the best possible hand is five of a kind Aces, or Template:Cards J. In casino lowball, the joker plays as the lowest card not already in the hand.

### LowballEdit

Main article: Lowball (poker)

Some games called lowball or low poker are played with the hand rankings the same but the objective reversed: players strive not for the highest ranking of the above hands but for the lowest ranking hand. There are three methods of ranking low hands, called ace-to-five low, deuce-to-seven low, and ace-to-six low, of which the ace-to-five method is most common. A variant within this category is high-low poker, in which the highest and lowest hands split the pot, with the highest hand taking any odd chips if the pot does not divide equally.

## References Edit

1. The general form of permutations is
$P(n, r) = \frac{n!}{(n-r)!}.$
thus: $P(52,5) = \frac {52!} {(52-5)!} = \frac {52!}{47!} = 311,875,200$
2. Template:Cite web
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