Number Of Unique Poker Hands

  
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The card game of poker has many variations, most of which were created in the United States in the mid-1900s. The standard order of play applies to most of these games, but to fully specify a poker game requires details about which hand values are used, the number of betting rounds, and exactly what cards are dealt and what other actions are taken between rounds.

Popular poker variants[edit]

A poker hand comprises five cards. The value of the hand is in inverse proportion to its mathematical frequency; that is, the more unusual the combination of cards, the higher the hand ranks. Players may bet that they have the best hand, and other players must either call (i.e., match) the bet or concede.

The three most popular poker variants are spread in casinos and poker rooms worldwide and can be divided into the following groups:

  • Draw poker: Games in which players are dealt a complete hand, hidden, and then improve it by replacing cards. The most common of these is five-card draw.
  • Stud poker: Games in which each player receives a combination of face-up cards and face-down cards in multiple betting rounds. The most common of these are five-card stud and seven-card stud. These two variants are further played in other different formats.
  • Community card poker: Games in which each player's incomplete hidden hand is combined with shared face-up cards. The most common of these are Texas hold 'em and Omaha hold 'em.

Here are more common rule variations:

  1. Lowball: The lowest hand wins the pot. There are different rules about whether or not aces count as low, and the effects of straights and flushes. The most common variants are Razz and 2-7 Triple Draw
  2. High-low split: the highest and lowest hands split the pot. Generally there is a qualifier for the low hand. For example, the low hand must have 5 cards with ranks of 8 or less. In most high-low games the usual rank of poker hands is observed, so that an unsuited broken straight (7-5-4-3-2) wins low (see Morehead, Official Rules of Card Games). In a variant, based on Lowball, where only the low hand wins, a straight or a flush does not matter for a low hand. So the best low hand is 5-4-3-2-A, suited or not.
  3. Players can pass cards to each other. An example of this would be Anaconda.
  4. Kill game: When a fixed limit game is played and a player wins two pots in a row, the stakes are doubled. In some split-pot games, a player winning both halves of the pot may also cause a kill. In some variants of Lowball, a player may choose to kill by placing a double bet after seeing his first two cards.
  5. Wild cards are added. This can range from simply making deuces wild to the 7-card stud variant named baseball.
  6. A twist round in which players can buy another card from the deck. If a player does not like the purchased card, the player can purchase another one by adding money to the pot. This is sometimes called a 'tittle.'
  7. A stripped deck may be used. Poker was first played with only 20 cards. In the spirit of poker history, players will sometimes only play with a stripped deck. A popular poker game in Spain is played with cards 8-A. It is played similar to hold'em, except that one card is dealt at a time and a player must use both hole cards.
  8. Roll your own is played in stud games, and allows the player to determine which of his or her cards are turned up and visible to the other players. In a game like Seven-card Stud, the 'roll' action only applies to the first 3 cards, all of which are dealt face down. Each player then determines which card to expose. Play then continues as with regular Seven-card Stud. But a game like Mexican stud applies the roll option throughout the game. Two cards are dealt face down, and the players roll one card up. The game continues just as in Five-card Stud, except the cards are dealt face down, and each player then decides which of the two down cards is exposed.[1]

Mixed poker games[edit]

Poker can be played in a mixed game format, for example one half-hour of Texas hold 'em followed by one half-hour of Seven-card stud. There are many types of mixed poker games. The most notable mixed poker variation is H.O.R.S.E. poker. H.O.R.S.E. is a mix of Texas hold 'em, Omaha high-low, Razz, Seven-card stud and Seven-card stud Eight-or-better. Each game will usually be played for a fixed number of hands or time and then the players will move on to the next game.

The poker odds calculators on CardPlayer.com let you run any scenario that you see at the poker table, see your odds and outs, and cover the math of winning and losing poker hands. Texas Hold'em Omaha. The hackerrank question asked me to write a program that would determine the best poker hand possible in five-card draw poker. We are given 10 cards, the first 5 are the current hand, and the second 5 are the next five cards in the deck. We assume that we can see the next five cards (they are not hidden).

Other mixed games include:

  • H.O.S.E. – same as H.O.R.S.E., except without Razz.
  • H.O.E. – same as H.O.R.S.E, except no Razz or Seven-card stud.
  • H.O. – Texas hold 'em and Omaha high-low
  • O.E. – Omaha high-low and Seven-card Stud Eight or better.
  • H.A. – Pot limit Texas hold 'em and pot limit Omaha.
  • Omaha High-low Mixed – Fixed limit Omaha and pot limit Omaha high-low
  • Holdem Mixed – Fixed limit and no limit Texas hold 'em.
  • Eight Game Mix – Fixed limit 2-7 Triple Draw, fixed limit Texas hold 'em, fixed limit Omaha Hi-Low Eight or better, Razz, fixed limit Seven-card stud, fixed limit Seven-card stud Hi-Low eight or better, no limit Texas hold 'em and pot limit Omaha.
  • Ten Game Mix (introduced at 2011 World Series of Poker) – No limit Hold'em, fixed limit Seven-card Razz, fixed limit Hold'em, fixed limit Badugi, fixed limit Seven-card Stud, no limit 2-7 Single Draw, fixed limit Omaha Hi-Low eight or better, pot limit Omaha, fixed limit 2-7 Triple Draw and fixed limit Seven-card stud Hi-Low eight or better (played with 6 players on a table).

Specific poker variant games[edit]

Some poker games don't fit neatly into the above categories, and some have features of more than one of these categories. These variants are most often played in home games, usually as part of a dealer's choice format.

High Chicago or Low Chicago[edit]

Either of these two versions can be played in any stud high game. In High Chicago, or sometimes simply called Chicago, the player with the highest spade face down (referred to as in the hole) receives half the pot. In Low Chicago, the player with the lowest spade in the hole receives half of the pot, with the A♠ being the lowest. If the player with the highest hand also has the highest/lowest spade in the hole, then that player receives the entire pot - having won both sides of the bet.

Number Of Unique Poker Hands Signs

Follow the Queen[edit]

This 7-card stud game uses a wild-card designated as whichever card is immediately dealt (exposed, or face-up) after any queen previously dealt (exposed). In the event that the final card dealt (exposed) is itself a queen, then all queens are wild. If no queens are dealt (exposed), then there are no wilds for that hand. Betting is the same as in normal 7-card stud games. Follow the Queen is a typical game variant in Dealer's Choice poker games.[2][3]

Countdown[edit]

In this 5-card game, the dealer gives each player 5 cards face down. After the first round of betting, each player may choose to replace zero to three cards. A second round of betting follows and then players may opt to replace zero to two cards. Upon completing another round of betting, each player may replace one card of the cards in their hand. After a final round of betting, any remaining players show their hands, and the highest 5 card hand wins. The cost of cards doubles each round. For example, if the dealer says each replacement card costs $10 in the first round, then each card costs $20 in the second round, and $40 in the final round. When players purchase cards to replace ones in their hand, they put the cost of the cards in the pot.[4]

This game will only work with 4 players (or fewer) otherwise you will run out of cards.

Billabong and Shanghai[edit]

Just as Oxford stud is a mixed stud/community card version of Texas hold 'em, Billabong is a mixed version of Manila. Each player is dealt two down-cards and one up-card. Low up-card starts the betting with a Bring-in if you are playing with one, otherwise high card starts the betting. Next, two community cards are dealt, followed by a second betting round, beginning with the player with the best exposed partial poker hand (counting the community cards, as in Oxford stud). Then a third community card is dealt, followed by a third betting round. Finally a fourth community card is dealt, followed by a fourth betting round and showdown. Each player plays the best five-card hand he can make from the three in his hand plus the four on the board in any combination.

Shanghai is the same game with an extra hole card, but no more than two hole cards play. That is, the game begins with each player being dealt three down-cards and one up-card; each player must discard one of his hole cards at some point during the game as determined ahead of time. The most common variation is to discard immediately as in Pineapple; the second most common is to discard just before showdown as in Tahoe.

Guts[edit]

Number Of Poker Hands

Guts is a family of games that are cousins of poker rather than poker variants. They usually involve hands of 3 or fewer cards, ranked similarly to hands in poker, and multiple successive rounds of betting each of which consist of the decision to be 'in' or 'out', and each with its own showdown. The losers of rounds of guts generally match or double the pot, which grows rapidly.

Five-O poker[edit]

Five-O Poker is a heads-up poker variant in which both players must play five hands of five cards simultaneously. Four of the five cards in each hand are face-up. Once all five hands are down, there is a single round of betting. The winner is determined by matching each hand to the corresponding hand of the opponent. The player with the stronger poker hand in three (or more) out of the five columns, wins, unless a player folds on a bet that was made. If a player beats their opponent with all five hands, this is called a “Five-O” win.

Chinese poker[edit]

Number Of Unique Poker Hands Held

Chinese poker is a 2-4 player poker game with thirteen cards. The idea is to make three poker hands with increasing rank : two with five cards and one with three cards. If one of the hands does not adhere to increasing rank (i.e. is mis-set), the hand is declared dead and results in some sort of penalty.

Kuhn poker[edit]

Kuhn poker, using a three card deck, is more of game theory problem than an actual game people play, but it can be played by two players.[5]

See also[edit]

References[edit]

  1. ^Pagat.com: Roll Your Own
  2. ^Pokernews.com: Fun Home Poker Game Rules - Follow the Queen
  3. ^Pokerrules.net: How to Play Follow the Queen
  4. ^Pokerrules.net: How to Play Countdown Poker
  5. ^'A Parameterized Family of Equilibrium Profiles for Three-Player Kuhn Poker'(PDF).
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One plays poker with a deck of 52 cards, which come in 4 suits (hearts, clubs, spades, diamonds) with 13 values per suit (A, 2, 3, …, 10, J, Q, K).

In poker one is dealt five cards and certain combinations of cards are deemed valuable. For example, a “four of a kind” consists of four cards of the same value and a fifth card of arbitrary value. A “full house” is a set of three cards of one value and two cards of a second value. A “flush” is a set of five cards of the same suit. The order in which one holds the cards in ones hand is immaterial.

EXAMPLE: How many flushes are possible in poker?

Answer: Again this is a multi-stage problem with each stage being its own separate labeling problem. One way to help tease apart stages is to image that you’ve been given the task of writing a computer program to create poker hands. How will you instruct the computer to create a flush?

First of all, there are four suits – hearts, spades, clubs and diamonds – and we need to choose one to use for our flush. That is, we need to label one suit as “used” and three suits as “not used.” There are (dfrac{4!}{1!3!} = 4) ways to do this.

Second stage: Now that we have a suit, we need to choose five cards from the 13 cards of that suit to use for our hand. Again, this is a labeling problem – label five cards as “used” and eight cards as “not used.” There are (dfrac{13!}{5!8!} = 1287) ways to do this.

By the multiplication principle there are (4 times 1287 = 5148) ways to compete both stages. That is, there are (5148) possible flushes.

COMMENT: There are (dfrac{52!}{5!47!} = 2598960) five-card hands in total in poker. (Why?) The chances of being dealt a flush are thus: (dfrac{5148}{2598960} approx 0.20%).

EXAMPLE: How many full houses are possible in poker?

Answer: This problem is really a three-stage labeling issue.

First we must select which of the thirteen card values – A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K – is going to be used for the triple, which will be used for the double, and which 11 values are going to be ignored. There are (dfrac{13!}{1!1!11!} = 13 times 12 = 156) ways to accomplish this task.

Among the four cards of the value selected for the triple, three will be used for the triple and one will be ignored. There are (dfrac{4!}{3!1!} = 4) ways to accomplish this task. Among the four cards of the value selected for the double, two will be used and two will be ignored. There are (dfrac{4!}{2!2!} = 6) ways to accomplish this.

By the multiplication principle, there are (156 times 4 times 6 = 3744) possible full houses.

COMMENT: High-school teacher Sam Miskin recently used this labeling method to count poker hands with his high-school students. To count how many “one pair hands” (that is, hands with one pair of cards the same numerical value and three remaining cards each of different value) he found it instructive bring 13 students to the front of the room and hand each student four cards the same value from a single deck of cards.

He then asked the remaining students to select which of the thirteen students should be the “pair” and which three should be the “singles.” He had the remaining nine students return to their seats.

He then asked the “pair” student to raise his four cards in the air and asked the seated students to select which two of the four should be used for the pair. He then asked each of the three “single” students in turn to hold up their cards while the seated students selected on one the four cards to make a singleton.

This process made the multi-stage procedure clear to all and the count of possible one pair hands, namely,

(dfrac{13!}{1!3!9!} times dfrac{4!}{2!2!} times 4 times 4 times 4)

readily apparent.

Exercise: “Two pair” consists of two cards of one value, two cards of a different value, and a third card of a third value. What are the chances of being dealt two-pair in poker?
EXAMPLE: A “straight” consists of five cards with values forming a string of five consecutive values (with no “wrap around”). For example, 45678, A2345 and 10JQKA are considered straights, but KQA23 is not. (Suits are immaterial for straights.)

How many different straights are there in poker?

Answer: A straight can begin with A, 2, 3, 4, 5, 6, 7, 8, 9 or 10. We must first select which of these values is to be the start of our straight. There are 10 choices.

For the starting value we must select which of the four suits it will be. There are 4 choices.

There are also 4 choices for the suit of the second card in the straight, 4 for the third, 4 for the fourth, and 4 for the fifth.

By the multiplication principle, the total number of straights is:

(10 times 4 times 4 times 4 times 4 times 4 = 10240).

Poker

The chances of being dealt a straight are about 0.39%.