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In poker, the probability of each type of 5 card hand can be computed by calculating the proportion of hands of that type among all possible hands.

Derivation

The following computations show how the above frequencies were determined. To understand these derivations, the reader should be familiar with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

• Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (T-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is: 40.
• Four of a kind — Any one of the thirteen ranks can form the four of a kind, leaving 52 − 4 = 48 possibilities for the final card. Thus, the total number of four-of-a-kinds is: 624
• Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and any three of the four suits. The pair can be any one of the remaining twelve ranks, and any two of the four suits. Thus, the total number of full houses is: 3,744
• Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is: 5,108
• Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-T. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving: 10,200
• Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The other cards can have any two of the remaining twelve ranks, and each can have any one of the four suits. Thus, the total number of three-of-a-kinds is: 54,912
• Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is: 123,552
• Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is: 1,098,240
• No pair — A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is: 1,302,540

Frequency of 7 card poker hands

In some popular variations of poker, a player uses the best five-card poker hand out of seven cards. The frequencies, probabilities, and odds are calculated as above; however the total numbers are greater since there are 133,784,560 (over 50 times more) 7 card combinations. It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand. (The frequencies given are exact; the probabilities and odds are approximate.)

Links

• MathWorld: Poker
• Probability and Poker
• Brian Alspach’s mathematics and poker page
• Brian Alspach Poker Math articles
• Brian Alspach Poker Math computations
• QA – Poker probabilities including conditional calculations
• 5, 6, and 7 card poker probabilities

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Video: PPA Poker Calculator, Win Probability

Gambling has had many different meanings depending on the cultural and historical context in which it is used. Currently, in western society, it generally has an economic definition and meaning and typically refers to “wagering money or something of material value on something with an uncertain outcome in hope of winning additional money or material goods”. Furthermore:

• the outcome of the wager is typically evident within a short period of time
• the primary intent of the bet is to win additional money or material goods

This definition of gambling usually excludes:

• emotional or physical risk-taking where what is being risked is not money or material goods (e.g., skydiving, running for office, asking someone for a date, etc.)
• buying insurance, as the primary intent of the purchase is to protect against loss, rather than to collect or win
• all forms of long-term ‘investment’ (stock market, real estate) with positive expected returns and economic utility
• starting a new business, as time and effort are also being wagered and the outcome is not determined in a short period of time
• situations where the possibility of winning additional money or material goods is a secondary or incidental reason for the wager/purchase (e.g., buying a raffle ticket to support a worthy cause)

Gambling varies on four dimensions:

• 1. What is being wagered (money or material goods).
• 2. How much is being wagered.
• 3. The predictability of the event. For some things such as lotteries, slot machines and bingo, the results are random and unpredictable. No skill or system will give you any advantage. For other things such as sports betting and horse racing there is some predictability to the outcome. In this situation greater knowledge and skill gives a person an advantage over other bettors.
• 4. The ‘expected value’, the positive or negative mathematical expectation.

Bibliography

• Brisman, Andrew. American Mensa Guide to Casino Gambling: Winning Ways (Stirling, 1999) ISBN 080694837X
• Ortiz, Darwin. Gambling Scams: How They Work, How to Detect Them, How to Protect Yourself (Carol, 1990) ISBN 0396083668 (Hardcover) ISBN 0818405295 (Paperback)
• Reith, Gerda. Age of Chance: Gambling in Western Culture ISBN 0415179971 (Hardcover) ISBN 0415263093 (Paperback)
• Steinmetz, Andrew. The Gaming Table: Its Votaries and Victims Gutenberg text
• Thorp, Edward O. Beat the Dealer: A Winning Strategy for the Game of Twenty-One (Vintage, 1966) ISBN 0394703103

Links

• DMOZ Gambling directory
• Yahoo Gambling directory
• Gaming Studies Research Center – at University of Nevada, Las Vegas
• DictionaryOfGambling.com
• United States Gambling Laws
• Institute for the Study of Gambling and Commercial Gaming at the University of Nevada, Reno
• Wizard of Odds – game descriptions, etiquette, and odds
• Gamblers Anonymous International

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Video: Gambling addiction and the teens

The standard poker hands in descending order.

A hand in poker can mean any of the following:

1. A unit of play consisting of a deal, one or more rounds of betting, and possibly a showdown.
2. A set of five cards with a certain value. For example, the hand A♥ 10♥ 9♥ 5♥ 3♥ is a “flush”, a hand that is valuable because each card is of the same suit.
3. A player’s set of non-communal cards.

The second and third definitions are often used interchangeably. For example, in Texas hold ‘em, a player holding A♣ K♠, with a board of A♥ K♣ K♦ 7♠ 3♦, might say, “my hand is ace-king”. However, his best 5-card hand (the portion of the hand which determines value) is the kings-over-aces full house.

General rules

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 (low).

Individual card ranks are often used to evaluate hands that contain no pairs or other special combinations, or to rank the kickers of otherwise equal hands. The Ace is ranked low in ace-to-five and ace-to-six lowball games.

• 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. Sometimes a ranking called high card by suit is used for randomly selecting a player to deal.

• A hand always consists of five cards.

In games where more than five cards are available to each player, hands are ranked by choosing some five-card subset according to the rules of the game, and comparing that five-card hand against the five-card hands of the other players. Whatever cards remain after choosing the five to be played are of no consequence in determining the winner. (For example, when comparing identical full houses, there are no “kickers”.)

• Hands are ranked first by category, then by individual card ranks.

That is, even the minimum qualifying hand in a certain category defeats all hands in all lower categories. The smallest Two pair hand, for example, defeats all hands with just One pair or No pair. Only between two hands in the same category are card ranks used to break ties. The highest single card in each flush or straight is used to break ties (the Ace-through-five straight is the lowest straight, the Ace being a low card in this context). Within two Two pair hands, the higher pairs are first compared. If they tie, then the secondary pairs are compared, and then finally the kicker.

• The order in which cards are dealt is unimportant.

For ease of explanation, hands are shown here neatly arranged, but a poker hand has the same value no matter what order the cards are received in.

Ranking of hands

The most common ranking of hands is as follows:

• Royal flush: Five cards in sequence and of the same suit, starting from the Ace down to the 10. Example: A♠ K♠ Q♠ J♠ 10♠ (Note: A Royal Flush is not a category of hand in and of itself, it is simply the highest-valued straight flush, and thus also the highest-valued hand. Since it is mentioned often in the context of hand rankings, it is worth noting in this list.)
• Straight flush: Any five cards in sequence and of the same suit. Example: Q♦ J♦ 10♦ 9♦ 8♦
• Four of a kind: A hand with four cards of the same rank. Example: 4♣ 4♦ 4♥ 4♠ 9♥
• Full house: A hand with three cards of one rank and two of another. Example: 8♣ 8♦ 8♠ K♥ K♠
• Flush: Five cards of the same suit. Example: K♠ J♠ 8♠ 4♠ 3♠
• Straight: Five cards in sequence. (The ace can be considered higher than the king, or lower than the two.) Example: 5♦ 4♥ 3♠ 2♦ A♦
• Three of a kind: Three cards of the same rank. Example: 7♣ 7♥ 7♠ K♦ 2♠
• Two pair: Two cards of one rank, two of another. Example: A♣ A♦ 8♥ 8♠ Q♠
• One pair: Two cards of the same rank. Example: 9♥ 9♠ A♣ J♠ 4♥
• No pair: Also known as a high card hand. The following example is considered “Ace high.” Example: A♦ 10♦ 9♠ 5♣ 4♣

The hands are ranked in this order because of their relative probabilities, with rarer hands ranking above more common hands.

An additional hand type, five of a kind, exists when wild cards are used. Five of a kind outranks the straight flush (and therefore the royal flush too) making it the most valuable hand.

Variations

Some games called lowball or low poker are played where players strive not for the highest ranking of the above combinations 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. The ace-to-five method is most common. A sub-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). Sometimes straights and/or flushes count in determining which hand is highest but not in determining which hand is lowest (being reckoned as a no-pair hand in the latter instance), so that a player with such a holding can win both ways and thus take the entire pot.

Certain variants use hands of only three cards, either high or low. Three-card low hands can be ranked by any of the three methods above, although with three cards they become ace-to-three (rather than ace-to-five), deuce-to-five, and ace-to-four. The ace-to-three method is the most common, just as the ace-to-five method is most common method for five cards. Three-card high hands are ranked in one of two ways: either with or without straights and flushes. Without them (which is the most common, and used such games as Chinese poker), the hands are simply no pair, one pair, and three of a kind. If you add straights and flushes, the order of hands should be changed to reflect the correct probabilities: no pair, one pair, flush, straight, three of a kind, straight flush. This order is used, for example, in Mambo stud.

Some poker games are played with a deck that has been stripped of certain cards, usually low-ranking ones. For example, the Australian game of Manila uses a 32-card deck in which all cards below the rank of 7 are removed, and Mexican stud removes the 8s, 9s, and 10s. In both of these games, a flush ranks above a full house, because having fewer cards of each suit available makes flushes rarer.

Some games add one or more unconventional hands, or have special exceptions to the rules above. For example, in the game of Pai gow poker as played in Nevada, a Wheel (5-4-3-2-A) ranks above a king-high straight, but below an ace-high straight. This is not the case in California, where the nearly identical game is played under the name Double-hand poker using traditional hand values.

Links

• Poker Hand Ranks

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Video: Poker hand rankings

Poker is a card game, the most popular of a class of games called vying games, in which players with fully or partially concealed cards make wagers into a central pot, which is awarded to the remaining player or players with the best combination of cards. Poker can also refer to video poker which is a single-player game seen in casinos much like a slot machine.

In order to play, one must learn the basic rules and procedures of the game, the values of the various combinations of cards , and the rules about betting limits. Some knowledge of the equipment used to play is useful. There are also many variants of poker, loosely categorized as draw poker, stud poker, community card poker (a.k.a. “widow game”), and miscellaneous poker games. The most commonly played games of the first three categories are five-card draw, seven-card stud, and Texas hold ‘em, respectively; each being a common starting point for learning games of the type. Dealer’s choice is a way to play poker where the dealer chooses what type of poker to play.

A game of Texas hold ‘em in progress. “Hold ‘em” is currently the most popular form of poker.

References

• Brunson, Doyle (1979). Doyle Brunson’s Super System, Cardoza. ISBN 1580420818.
• Sklansky, David (1989). The Theory of Poker (3rd Ed), Two Plus Two Publications. ISBN 1880685000.
• Vorhaus, John (2002). Killer Poker, Lyle Stuart. ISBN 0818406305.
• Ernest, James; Selinker, Mike; Foglio, Phil (2005). Dealer’s Choice: The Complete Handbook of Saturday Night Poker, Overlook Press. ISBN 1585676543.
• Caro, Mike (1978). Caro’s Book of Poker Tells, Cardoza. ISBN 1580420826.

Links

• Yahoo Poker Directory
• Google Poker Directory
• Rec.Gambling.Poker Newsgroup

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.