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.
Probability and gambling have been ideas since long before the invention of poker. The development of probability theory in the late 1400s was attributed to gambling; when playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability. Motivated by Paccioli's work, Girolamo Cardano (1501-1576) made further developments in probability theory. His work from 1550, titled Liber de Ludo Aleae, discussed the concepts of probability and how they were directly related to gambling. However, his work did not receive any immediate recognition since it was not published until after his death. Blaise Pascal (1623-1662) also contributed to probability theory. His friend, Chevalier de Méré, was an avid gambler with the goal to become wealthy from it. De Méré tried a new mathematical approach to a gambling game but did not get the desired results. Determined to know why his strategy was unsuccessful, he consulted with Pascal. Pascal's work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat (1601-1665). Communicating through letters, the two continued to exchange their ideas and thoughts. These interactions led to the conception of basic probability theory. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. [1] [2]
In straight poker and five-card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52.
The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. In this chart:
The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr
with 52
and 5
, for example, yields as above.
Hand | Distinct hands | Frequency | Probability | Cumulative probability | Odds against | Mathematical expression of absolute frequency |
---|---|---|---|---|---|---|
Royal flush | 1 | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
Straight flush (excluding royal flush) | 9 | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
Four of a kind | 156 | 624 | 0.02401% | 0.0255% | 4,164 : 1 | |
Full house | 156 | 3,744 | 0.1441% | 0.17% | 693.1667 : 1 | |
Flush (excluding royal flush and straight flush) | 1,277 | 5,108 | 0.1965% | 0.37% | 507.8019 : 1 | |
Straight (excluding royal flush and straight flush) | 10 | 10,200 | 0.3925% | 0.76% | 253.8 : 1 | |
Three of a kind | 858 | 54,912 | 2.1128% | 2.87% | 46.32955 : 1 | |
Two pair | 858 | 123,552 | 4.7539% | 7.63% | 20.03535 : 1 | |
One pair | 2,860 | 1,098,240 | 42.2569% | 49.9% | 1.366477 : 1 | |
No pair / High card | 1,277 | 1,302,540 | 50.1177% | 100% | 0.9953015 : 1 | |
Total | 7,462 | 2,598,960 | 100% | --- | 0 : 1 |
The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.
Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.
The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.
In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall, [3] a player uses the best five-card poker hand out of seven cards.
The frequencies are calculated in a manner similar to that shown for 5-card hands, [4] except additional complications arise due to the extra two cards in the 7-card poker hand. The total number of distinct 7-card hands is . It is notable that the probability of a no-pair hand is lower than the probability of a one-pair or two-pair hand.
The Ace-high straight flush or royal flush is slightly more frequent (4324) than the lower straight flushes (4140 each) because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace of its suit in the hand (as that would make it ace-high instead).
Hand | Frequency | Probability | Cumulative | Odds against | Mathematical expression of absolute frequency |
---|---|---|---|---|---|
Royal flush | 4,324 | 0.0032% | 0.0032% | 30,939 : 1 | |
Straight flush (excluding royal flush) | 37,260 | 0.0279% | 0.0311% | 3,589.6 : 1 | |
Four of a kind | 224,848 | 0.168% | 0.199% | 594 : 1 | |
Full house | 3,473,184 | 2.60% | 2.80% | 37.5 : 1 | |
Flush (excluding royal flush and straight flush) | 4,047,644 | 3.03% | 5.82% | 32.1 : 1 | |
Straight (excluding royal flush and straight flush) | 6,180,020 | 4.62% | 10.4% | 20.6 : 1 | |
Three of a kind | 6,461,620 | 4.83% | 15.3% | 19.7 : 1 | |
Two pair | 31,433,400 | 23.5% | 38.8% | 3.26 : 1 | |
One pair | 58,627,800 | 43.8% | 82.6% | 1.28 : 1 | |
No pair / High card | 23,294,460 | 17.4% | 100% | 4.74 : 1 | |
Total | 133,784,560 | 100% | --- | 0 : 1 |
(The frequencies given are exact; the probabilities and odds are approximate.)
Since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands.
The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high).
Some variants of poker, called lowball, use a low hand to determine the winning hand. In most variants of lowball, the ace is counted as the lowest card and straights and flushes don't count against a low hand, so the lowest hand is the five-high hand A-2-3-4-5, also called a wheel. The probability is calculated based on , the total number of 5-card combinations. (The frequencies given are exact; the probabilities and odds are approximate.)
Hand | Distinct hands | Frequency | Probability | Cumulative | Odds against |
---|---|---|---|---|---|
5-high | 1 | 1,024 | 0.0394% | 0.0394% | 2,537.05 : 1 |
6-high | 5 | 5,120 | 0.197% | 0.236% | 506.61 : 1 |
7-high | 15 | 15,360 | 0.591% | 0.827% | 168.20 : 1 |
8-high | 35 | 35,840 | 1.38% | 2.21% | 71.52 : 1 |
9-high | 70 | 71,680 | 2.76% | 4.96% | 35.26 : 1 |
10-high | 126 | 129,024 | 4.96% | 9.93% | 19.14 : 1 |
Jack-high | 210 | 215,040 | 8.27% | 18.2% | 11.09 : 1 |
Queen-high | 330 | 337,920 | 13.0% | 31.2% | 6.69 : 1 |
King-high | 495 | 506,880 | 19.5% | 50.7% | 4.13 : 1 |
Total | 1,287 | 1,317,888 | 50.7% | 50.7% | 0.97 : 1 |
As can be seen from the table, just over half the time a player gets a hand that has no pairs, threes- or fours-of-a-kind. (50.7%)
If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.
Some players do not ignore straights and flushes when computing the low hand in lowball. In this case, the lowest hand is A-2-3-4-6 with at least two suits. Probabilities are adjusted in the above table such that "5-high" is not listed", "6-high" has one distinct hand, and "King-high" having 330 distinct hands, respectively. The Total line also needs adjusting.
In some variants of poker a player uses the best five-card low hand selected from seven cards. In most variants of lowball, the ace is counted as the lowest card and straights and flushes don't count against a low hand, so the lowest hand is the five-high hand A-2-3-4-5, also called a wheel. The probability is calculated based on , the total number of 7-card combinations.
The table does not extend to include five-card hands with at least one pair. Its "Total" represents the 95.4% of the time that a player can select a 5-card low hand without any pair.
Hand | Frequency | Probability | Cumulative | Odds against |
---|---|---|---|---|
5-high | 781,824 | 0.584% | 0.584% | 170.12 : 1 |
6-high | 3,151,360 | 2.36% | 2.94% | 41.45 : 1 |
7-high | 7,426,560 | 5.55% | 8.49% | 17.01 : 1 |
8-high | 13,171,200 | 9.85% | 18.3% | 9.16 : 1 |
9-high | 19,174,400 | 14.3% | 32.7% | 5.98 : 1 |
10-high | 23,675,904 | 17.7% | 50.4% | 4.65 : 1 |
Jack-high | 24,837,120 | 18.6% | 68.9% | 4.39 : 1 |
Queen-high | 21,457,920 | 16.0% | 85.0% | 5.23 : 1 |
King-high | 13,939,200 | 10.4% | 95.4% | 8.60 : 1 |
Total | 127,615,488 | 95.4% | 95.4% | 0.05 : 1 |
(The frequencies given are exact; the probabilities and odds are approximate.)
If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand.
Some players do not ignore straights and flushes when computing the low hand in lowball. In this case, the lowest hand is A-2-3-4-6 with at least two suits. Probabilities are adjusted in the above table such that "5-high" is not listed, "6-high" has 781,824 distinct hands, and "King-high" has 21,457,920 distinct hands, respectively. The Total line also needs adjusting.
A wild card in card games is one that may be used to represent any other playing card, sometimes with certain restrictions. Jokers are often used as wild cards, but other cards may be designated as wild by the rules or by agreement. In addition to their use in card games played with a standard pack, wild cards may also exist in dedicated deck card games, such as the 'Master' card in Lexicon.
In poker, the nut hand is the strongest possible hand in a given situation. The second-nut hand or third-nut hand may refer to the second and third best possible hands. The term applies mostly to community card poker games where the individual holding the strongest possible hand, with the given board of community cards, is capable of knowing that they have the nut hand.
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to a flush. In contrast, a made hand already has value and does not necessarily need to draw to win. A made starting hand with no help can lose to an inferior starting hand with a favorable draw. If an opponent has a made hand that will beat the player's draw, then the player is drawing dead; even if they make their desired hand, they will lose. Not only draws benefit from additional cards; many made hands can be improved by catching an out – and may have to in order to win.
In a poker game with more than one betting round, an out is any unseen card that, if drawn, will improve a player's hand to one that is likely to win. Knowing the number of outs a player has is an important part of poker strategy. For example, in draw poker, a hand with four diamonds has nine outs to make a flush: there are 13 diamonds in the deck, and four of them have been seen. If a player has two small pairs, and he believes that it will be necessary for him to make a full house to win, then he has four outs: the two remaining cards of each rank that he holds.
Omaha hold 'em is a community card poker game similar to Texas hold 'em, where each player is dealt four cards and must make their best hand using exactly two of them, plus exactly three of the five community cards. The exact origin of the game is unknown, but casino executive Robert Turner first brought Omaha into a casino setting when he introduced the game to Bill Boyd, who offered it as a game at the Las Vegas Golden Nugget Casino. Omaha uses a 52-card French deck. Omaha hold 'em 8-or-better is the "O" game featured in H.O.R.S.E.
Seven-card stud, also known as Seven-Toed Pete or Down-The-River, is a variant of stud poker. Before the 2000s surge of popularity of Texas hold 'em, seven-card stud was one of the most widely played poker variants in home games across the United States and in casinos in the eastern part of the country. Although seven-card stud is not as common in casinos today, it is still played online. The game is commonly played with two to eight players, however, eight may require special rules for the last cards dealt if no players fold. Playing with nine players is possible.
Texas hold 'em is one of the most popular variants of the card game of poker. Two cards, known as hole cards, are dealt face down to each player, and then five community cards are dealt face up in three stages. The stages consist of a series of three cards, later an additional single card, and a final card. Each player seeks the best five-card poker hand from any combination of the seven cards: the five community cards and their two hole cards. Players have betting options to check, call, raise, or fold. Rounds of betting take place before the flop is dealt and after each subsequent deal. The player who has the best hand and has not folded by the end of all betting rounds wins all of the money bet for the hand, known as the pot. In certain situations, a "split pot" or "tie" can occur when two players have hands of equivalent value. This is also called "chop the pot". Texas hold 'em is also the H game featured in HORSE and HOSE.
Lowball or low poker is a variant of poker in which the normal ranking of hands is inverted. Several variations of lowball poker exist, differing in whether aces are treated as high cards or low cards, and whether straights and flushes are used.
Suited connectors is a poker term referring to pocket cards that are suited and consecutive, for example Q♥ J♥. These hands are considered stronger than average because they have the highest potential to form straights and flushes when combined with the community cards.
Non-standard poker hands are hands which are not recognized by official poker rules but are made by house rules. Non-standard hands usually appear in games using wild cards or bugs. Other terms for nonstandard hands are special hands or freak hands. Because the hands are defined by house rules, the composition and ranking of these hands is subject to variation. Any player participating in a game with non-standard hands should be sure to determine the exact rules of the game before play begins.
Razz is a form of stud poker that is normally played for ace-to-five low. It is one of the oldest forms of poker, and has been played since the start of the 20th century. It emerged around the time people started using the 52-card deck instead of 20 for poker.
Tiến lên is a shedding-type card game originating in Vietnam. It may be considered Vietnam's national card game, and is also played in the United States, sometimes under the names Viet Cong, VC, Thirteen, Killer, or 2’s.
Badugi is a draw poker variant similar to triple draw, with hand-values similar to lowball. The betting structure and overall play of the game is identical to a standard poker game using blinds, but, unlike traditional poker which involves a minimum of five cards, players' hands contain only four cards at any one time. During each of three drawing rounds, players can trade zero to four cards from their hands for new ones from the deck, in an attempt to form the best badugi hand and win the pot. Badugi is often a gambling game, with the object being to win money in the form of pots. The winner of the pot is the person with the best badugi hand at the conclusion of play. Badugi is played in cardrooms around the world, as well as online, in rooms such as PokerStars. Although it hasn’t had its own tournament per se at the WSOP, it is featured in the Dealers Choice events as well as in the Triple Draw Mix. The 2023 WSOP event does have a Badugi tournament scheduled.
Danny Nguyen is a poker dealer and poker player from San Jose, California.
Poker hand A is said to dominate poker hand B if poker hand B has three or fewer outs that would improve it enough to win. Informally, domination is sometimes used to refer to any situation where one hand is highly likely to beat another. The term drawing dead is used to denote a domination situation with zero outs.
In the game of contract bridge Fantunes is a natural bidding system initially developed Carlos Mosca and popularized by Fulvio Fantoni and Claudio Nunes - two players that were later found to be cheating and were banned from play by many bridge federations
Poker dice are dice which, instead of having number pips, have representations of playing cards upon them. Poker dice have six sides, one each of an Ace, King, Queen, Jack, 10, and 9, and are used to form a poker hand.
Baduci is a combination of Badugi poker and deuce to seven triple draw, and uses hand values similar to lowball. The pot in this game is split much like high-low split between the best Badugi hand and the best 2-7 triple draw hand. The betting structure and overall play of the game is nearly identical to a standard poker game using blinds. A players' hand contains five cards, where only four cards are used to determine the best Badugi hand and five cards are used to determine the triple draw hand. During each of three drawing rounds, players can trade zero to five cards from their hands for new ones from the deck.
Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or "drawing", cards.