HOME

TheInfoList



OR:

In
poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game w ...
, the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of each type of 5-card
hand A hand is a prehensile, multi-fingered appendage located at the end of the forearm or forelimb of primates such as humans, chimpanzees, monkeys, and lemurs. A few other vertebrates such as the koala (which has two opposable thumbs on each "h ...
can be computed by calculating the proportion of hands of that type among all possible hands.


History

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 Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
; 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 which was the first written text on probability. Motivated by Paccioli's work,
Girolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
(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 Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
(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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
(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.


Frequencies


5-card poker hands

In straight poker and
five-card draw Five-card draw (also known as a Cantrell draw) is a poker variant that is considered the simplest variant of poker, and is the basis for video poker. As a result, it is often the first variant learned by new players. It is commonly played in ho ...
, where there are no
hole card Community card poker refers to any game of poker that uses community cards (also called "shared cards"), which are cards dealt face up in the center of the table and shared by all players. In these games, each player is dealt an incomplete hand ...
s, players are simply dealt five cards from a deck of 52. The following chart enumerates the (absolute)
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of each hand, given all
combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
s of five cards
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
ly drawn from a full deck of 52 without replacement.
Wild cards ''Wild Cards'' is a series of science fiction superhero shared universe anthologies, mosaic novels, and solo novels. They are written by a collection of more than forty authors (referred to as the "Wild Cards Trust") and are edited by George R. ...
are not considered. In this chart: *Distinct hands is the number of different ways to draw the hand, not counting different suits. *Frequency is the number of ways to draw the hand, ''including'' the same card values in different suits. *The
Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
; = 2,598,960). For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. One would then expect to draw this hand about once in every 649,740 draws, or nearly 0.00000154% of the time. *Cumulative probability refers to the probability of drawing a hand as good as ''or better than'' the specified one. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand ''at least'' as good as three of a kind is about 2.87%. The cumulative probability is determined by adding one hand's probability with the probabilities of all hands above it. *The
Odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
are defined as the ratio of the number of ways ''not'' to draw the hand, to the number of ways to draw it. In statistics, this is called odds against. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. The formula for establishing the odds can also be stated as ''(1/p) - 1 : 1'', where ''p'' is the aforementioned probability. *The values given for Probability, Cumulative probability, and Odds are rounded off for simplicity; the Distinct hands and Frequency values are exact. The '' nCr'' function on most scientific calculators can be used to calculate hand frequencies; entering with and , for example, yields = 2,598,960 as above. 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.


7-card poker hands

In some popular variations of poker such as
Texas hold 'em Texas hold 'em (also known as Texas holdem, hold 'em, and holdem) 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 card poker, communit ...
, the most widespread poker variant overall′, 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,https://www.pokerstrategy.com/strategy/various-poker/texas-holdem-probabilities/ 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 = 133,784,560. 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). : (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).


5-card lowball poker hands

Some variants of poker, called lowball, use a
low hand 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. Lo ...
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 = 2,598,960, the total number of 5-card combinations. (The frequencies given are exact; the probabilities and odds are approximate.) : 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.


7-card lowball poker hands

In some variants of poker a player uses the best five-card
low hand 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. Lo ...
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 = 133,784,560, 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. : (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.


See also

*
Binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
*
Combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
*
Combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...
*
Effective hand strength algorithm Effective Hand Strength (EHS) is a poker algorithm conceived by computer scientists Darse Billings, Denis Papp, Jonathan Schaeffer and Duane Szafron that was published for the first time in the It has since then been considered as a reference ...
*
Event (probability theory) In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usua ...
*
Game complexity Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. Measures of game comple ...
*
Gaming mathematics Experiments, events and probability spaces The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples: * Throwing the dice in craps is an experiment that generates events such as occurrences of cer ...
*
Odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
*
Permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
*
Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
*
Sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
*
Set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...


References


External links


Brian Alspach's mathematics and poker page



Poker probabilities including conditional calculations






{{DEFAULTSORT:Poker Probability