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.

This is the highest poker hand. It consists of ace, king, queen, jack and ten, all in the. The strongest poker hand is the royal flush. It consists of Ten, Jack, Queen, King, and Ace, all of the same suit, e.g. Diamonds, spades, hearts, or clubs. #2 Straight Flush The second strongest hand in. 8, 7, 6, 5, 4 is the lowest of the low ranking hands. Poker Hand Probabilities. As you may have already figured out, the likelihood of making any of these hands increases the farther down the list you go – for the most part. A Royal Flush is very rare indeed, while a Pair happens quite a. Next in the poker hands list is a straight, consisting of a run of five cards of consecutive values, such as 4-5-6-7-8. Aces count as high or low, so you can make a 10-J-Q-K-A straight, the highest, or an A-2-3-4-5 straight, which is the lowest and sometimes called a “wheel”.

Popular poker variants[edit]

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

524230 players in the All Time Money List. This ranking list does not include results from recurring events (regular daily, weekly or monthly events).

• 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:

Poker Hands Chart

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.

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]

List Of Poker Hands Pdf

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.

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]

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]

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).

This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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What Are The Different Types Of Poker Hands

Preliminary Calculation

Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

These are the same hand. Order is not important.

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.

Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is

This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.

The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.

If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.

Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.

Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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The Poker Hands

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

Definitions of Poker Hands

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Counting Poker Hands

Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

Two Pair and One Pair
These two are left as exercises.

High Card
The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

Probabilities of Poker Hands

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2017 – Dan Ma