Editing Poker probability

{{Short description|Chances of card combinations in poker}}
In [[poker]], the [[probability]] of each type of 5-card [[poker hand|hand]] 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]]; when playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra [[Luca Pacioli]] released his work {{lang|la|Summa de arithmetica, geometria, proportioni e proportionalita}} which was the first written text on probability. Motivated by Pacioli'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.<ref>{{cite web|title=Probability Theory|url=http://www.scienceclarified.com/Ph-Py/Probability-Theory.html|website=Science Clarified|access-date=7 December 2015}}</ref><ref>{{cite web|title=Brief History of Probability|url=http://www.teacherlink.org/content/math/interactive/probability/history/briefhistory/home.html|website=teacher link|access-date=7 December 2015}}</ref>

==Frequencies==
===5-card poker hands===
[[File:Video poker JoB odds.svg|thumb|An [[Euler diagram]] depicting poker hands and their odds from a typical American 9/6 [[Video poker#Jacks or Better|Jacks or Better]] machine]]

In [[straight poker]] and [[five-card draw]], where there are no [[hole card]]s, players are simply dealt five cards from a deck of 52.

The following chart [[enumeration|enumerates]] the (absolute) [[frequency]] of each hand, given all [[combination]]s of five cards [[random]]ly drawn from a full deck of 52 without replacement. [[Wild Card (card games)|Wild cards]] are not considered. In this chart:

*'''Distinct hands''' is the number of different ways to draw the hand, not counting different suits.  In particular, a set of hands that all tie each other is counted exactly once, not multiply.
*'''Frequency''' is the number of ways to draw the hand, ''including'' the same card values in different suits.
*The '''[[Probability]]''' 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]]; <math display="inline">{52 \choose 5} = 2,598,960</math>). For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is {{sfrac|4|2,598,960}}, or one in 649,740. One would then expect to draw this hand about once in every 649,740 draws, or nearly 0.000154% 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]]''' 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 ''[[Binomial coefficient|nCr]]'' or ''[[Binomial coefficient|nCk]]'' function on most scientific calculators can be used to calculate hand frequencies; entering {{code|nCr}} with {{code|52}} and {{code|5}}, for example, yields <math display="inline">{52 \choose 5} = 2,598,960</math> as above.

{| class="wikitable" style="text-align:center;"
|-
! Hand !! Distinct hands !! Frequency !! Probability !! Cumulative probability !! Odds against
! Mathematical expression of absolute frequency
|-
| [[Hand rankings#Straight flush|Royal flush]]<br />
{{card|spade|10|40px}} {{card|spade|J|40px}} {{card|spade|Q|40px}} {{card|spade|K|40px}} {{card|spade|A|40px}}
| 1
| 4
| 0.000154%
| 0.000154%
| 649,739 : 1
| <math>{4 \choose 1}</math>
|-
| [[Hand rankings#Straight flush|Straight flush]] (excluding royal flush)<br />
{{card|heart|4|40px}} {{card|heart|5|40px}} {{card|heart|6|40px}} {{card|heart|7|40px}} {{card|heart|8|40px}}
| 9
| 36
| 0.00139%
| 0.00154%
| 72,192.33 : 1
| <math>{10 \choose 1}{4 \choose 1} - {4\choose 1}</math>
|-
| [[Hand rankings#Four of a kind|Four of a kind]]<br />
{{card|heart|A|40px}} {{card|diamond|A|40px}} {{card|club|A|40px}} {{card|spade|A|40px}} {{card|diamond|4|40px}}
| 156
| 624
| 0.02401%
| 0.0255%
| 4,164 : 1
| <math>{13 \choose 1}{4 \choose 4}{12 \choose 1}{4 \choose 1}</math>
|-
| [[Hand rankings#Full house|Full house]]<br />
{{card|heart|8|40px}} {{card|diamond|8|40px}} {{card|club|8|40px}} {{card|heart|K|40px}} {{card|spade|K|40px}}
| 156
| 3,744
| 0.1441%
| 0.17%
| 693.1667 : 1
| <math>{13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2}</math>
|-
| [[Hand rankings#Flush|Flush]] (excluding royal flush and straight flush)<br />
{{card|club|10|40px}} {{card|club|4|40px}} {{card|club|Q|40px}} {{card|club|7|40px}} {{card|club|2|40px}}
| 1,277
| 5,108
| 0.1965%
| 0.37%
| 507.8019 : 1
| <math>{13 \choose 5}{4 \choose 1} - {10 \choose 1}{4 \choose 1}</math>
|-
| [[Hand rankings#Straight|Straight]] (excluding royal flush and straight flush)<br />
{{card|club|7|40px}} {{card|heart|8|40px}} {{card|diamond|9|40px}} {{card|heart|10|40px}} {{card|spade|J|40px}}
| 10
| 10,200
| 0.3925%
| 0.76%
| 253.8 : 1
| <math>{10 \choose 1}{4 \choose 1}^5 - {10 \choose 1}{4 \choose 1}</math>
|-
| [[Hand rankings#Three of a kind|Three of a kind]]<br />
{{card|heart|Q|40px}} {{card|club|Q|40px}} {{card|diamond|Q|40px}} {{card|spade|5|40px}} {{card|diamond|A|40px}}
| 858
| 54,912
| 2.1128%
| 2.87%
| 46.32955 : 1
| <math>{13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2</math>
|-
| [[Hand rankings#Two pair|Two pair]]<br />
{{card|heart|3|40px}} {{card|diamond|3|40px}} {{card|club|6|40px}} {{card|heart|6|40px}} {{card|spade|K|40px}}
| 858
| 123,552
| 4.7539%
| 7.63%
| 20.03535 : 1
| <math>{13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1}</math>
|-
| [[Hand rankings#One pair|One pair]]<br />
{{card|heart|5|40px}} {{card|spade|5|40px}} {{card|club|2|40px}} {{card|club|J|40px}} {{card|diamond|A|40px}}
| 2,860
| 1,098,240
| 42.2569%
| 49.9%
| 1.366477 : 1
| <math>{13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3</math>
|-
| [[Hand rankings#High card|No pair]] / High card<br />
{{card|diamond|2|40px}} {{card|spade|5|40px}} {{card|spade|6|40px}} {{card|heart|J|40px}} {{card|club|A|40px}}
| 1,277
| 1,302,540
| 50.1177%
| 100%
| 0.9953015 : 1
| <math>\left[{13 \choose 5} - {10 \choose 1}\right] \left[{4 \choose 1}^5 - {4 \choose 1}\right]</math>
|-
! Total
! 7,462
! 2,598,960
! 100%
! ---
! 0 : 1
! <math>{52 \choose 5}</math>
|}

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 '''<span style="color:red;">3♦ 7♦ 8♦ Q♥ A♥</span>''' 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 '''<span style="color:red;">3♦</span> 7♣ <span style="color:red;">8♦ Q♥ A♥</span>''' 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]], the most widespread poker variant overall,<ref>{{cite web | url=https://www.casinodaniabeach.com/most-popular-types-of-poker/ | title=How to Play the Most Popular Types of Poker | date=14 August 2019 }}</ref> 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,<ref>{{cite web | url=https://www.pokerstrategy.com/strategy/various-poker/texas-holdem-probabilities/ | title=Probabilities in Texas Hold'em }}</ref> 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 <math display="inline">{52 \choose 7} = 133{,}784{,}560</math>.  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 [[Playing card|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).

:{| class="wikitable" style="font-size: 95%; text-align:center;"
|-
!Hand         !! Frequency !! Probability !! Cumulative !! Odds against
!Mathematical expression of absolute frequency
|-
|[[Hand rankings#Straight flush|Royal flush]]<br />
{{card|spade|10|40px}} {{card|spade|J|40px}} {{card|spade|Q|40px}} {{card|spade|K|40px}} {{card|spade|A|40px}}
|align=center|        4,324 ||   0.0032%   ||   0.0032%  ||align=center| 30,939 : 1
|<math>{4 \choose 1}{47 \choose 2}</math>
|-
|[[Hand rankings#Straight flush|Straight flush]] (excluding royal flush)<br />
{{card|heart|4|40px}} {{card|heart|5|40px}} {{card|heart|6|40px}} {{card|heart|7|40px}} {{card|heart|8|40px}}
|align=center|       37,260 ||   0.0279%   ||   0.0311%  ||align=center| 3,589.6 : 1
|<math>{9 \choose 1}{4 \choose 1}{46 \choose 2}</math>
|-
|[[Hand rankings#Four of a kind|Four of a kind]]<br />
{{card|heart|A|40px}} {{card|diamond|A|40px}} {{card|club|A|40px}} {{card|spade|A|40px}} {{card|diamond|4|40px}}
|align=center|      224,848 ||   0.168%    ||   0.199%   ||align=center| 594 : 1
|<math>{13 \choose 1}{48 \choose 3}</math>
|-
|[[Hand rankings#Full house|Full house]]<br />
{{card|heart|8|40px}} {{card|diamond|8|40px}} {{card|club|8|40px}} {{card|heart|K|40px}} {{card|spade|K|40px}}
|align=center|    3,473,184 ||   2.60%     ||   2.80%    ||align=center| 37.5 : 1
|<math>\begin{align}
& \left[ {13 \choose 2}{4 \choose 3}^2{44 \choose 1} \right] \\
+ & \left[{13 \choose 1}{12 \choose 2}{4 \choose 3}{4 \choose 2}^2 \right] \\
+ & \left[{13 \choose 1}{12 \choose 1}{11 \choose 2}{4 \choose 3}{4 \choose 2}{4 \choose 1}^2\right]
\end{align}</math>
|-
|[[Hand rankings#Flush|Flush]] (excluding royal flush and straight flush)<br />
{{card|club|10|40px}} {{card|club|4|40px}} {{card|club|Q|40px}} {{card|club|7|40px}} {{card|club|2|40px}}
|align=center|    4,047,644 ||   3.03%     ||   5.82%    ||align=center| 32.1 : 1
|<math>\begin{align}
& \left[ {4 \choose 1} \times \left[ {13 \choose 7} - 217\right] \right] \\
+ & \left[ {4 \choose 1} \times \left[ {13 \choose 6} - 71\right] \times 39\right] \\
+ & \left[ {4 \choose 1} \times \left[ {13 \choose 5} - 10\right] \times {39 \choose 2}\right]
\end{align}</math>
|-
|[[Hand rankings#Straight|Straight]] (excluding royal flush, straight flush, and overlapping flushes)<br />
{{card|club|7|40px}} {{card|heart|8|40px}} {{card|diamond|9|40px}} {{card|heart|10|40px}} {{card|spade|J|40px}}
|align=center|    6,180,020 ||   4.62%     ||  10.4%     ||align=center| 20.6 : 1
|<math>\begin{align}
& \left[ 217 \times \left[4^7 - 756 - 4 - 84\right] \right] \\
+ &{} \left[ 71 \times 36 \times 990 \right] \\
+ & \left[ 10 \times 5 \times 4 \times \left[256 - 3\right] + 10 \times {5 \choose 2} \times 2268 \right]
\end{align}</math>
|-
|[[Hand rankings#Three of a kind|Three of a kind]]<br />
{{card|heart|Q|40px}} {{card|club|Q|40px}} {{card|diamond|Q|40px}} {{card|spade|5|40px}} {{card|diamond|A|40px}}
|align=center|    6,461,620 ||   4.83%     ||  15.3%     ||align=center| 19.7 : 1
|<math>\left[{13 \choose 5} - 10\right]{5 \choose 1}{4 \choose 1}\left[{4 \choose 1}^4 - 3\right]</math>
|-
|[[Hand rankings#Two pair|Two pair]]<br />
{{card|heart|3|40px}} {{card|diamond|3|40px}} {{card|club|6|40px}} {{card|heart|6|40px}} {{card|spade|K|40px}}
|align=center|   31,433,400 ||  23.5%      ||  38.8%     ||align=center| 3.26 : 1
|<math>\begin{align}
& \left[ 1277 \times 10 \times \left[6 \times 62 + 24 \times 63 + 6 \times 64\right] \right] \\
+ & \left[ {13 \choose 3} {4 \choose 2} ^ 3 {40 \choose 1}\right]
\end{align}</math>
|-
|[[Hand rankings#One pair|One pair]]<br />
{{card|heart|5|40px}} {{card|spade|5|40px}} {{card|club|2|40px}} {{card|club|J|40px}} {{card|diamond|A|40px}}
|align=center|   58,627,800 ||  43.8%      ||  82.6%     ||align=center| 1.28 : 1
|<math>\left[{13 \choose 6} - 71\right] \times 6 \times 6 \times 990</math>
|-
|[[Hand rankings#High card|No pair]] / High card<br />
{{card|diamond|2|40px}} {{card|spade|5|40px}} {{card|spade|6|40px}} {{card|heart|J|40px}} {{card|club|A|40px}}
|align=center|   23,294,460 ||  17.4%      || 100%       ||align=center| 4.74 : 1
|<math>1499 \times \left[4^7 - 756 - 4 - 84\right]</math>
|-
! Total
! align="center" | 133,784,560 !! 100% !! --- !! align="center" | 0 : 1
!<math>{52 \choose 7}</math>
|}

(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===
{{See also|Lowball (poker)}}

Some variants of poker, called [[Lowball (poker)|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 <math display="inline">{52 \choose 5} = 2,598,960</math>, the total number of 5-card combinations.  (The frequencies given are exact; the probabilities and odds are approximate.)

:{| class="wikitable" style="font-size: 95%;"
|-
!Hand        !!   Distinct hands !!            Frequency !! Probability !! Cumulative !! Odds against
|-
|     5-high ||align=right|    1 ||align=right|    1,024 ||     0.0394% ||    0.0394% ||align=right| 2,537.05 : 1
|-
|     6-high ||align=right|    5 ||align=right|    5,120 ||     0.197%  ||    0.236%  ||align=right| 506.61 : 1
|-
|     7-high ||align=right|   15 ||align=right|   15,360 ||     0.591%  ||    0.827%  ||align=right| 168.20 : 1
|-
|     8-high ||align=right|   35 ||align=right|   35,840 ||     1.38%   ||    2.21%   ||align=right| 71.52 : 1
|-
|     9-high ||align=right|   70 ||align=right|   71,680 ||     2.76%   ||    4.96%   ||align=right| 35.26 : 1
|-
|    10-high ||align=right|  126 ||align=right|  129,024 ||     4.96%   ||    9.93%   ||align=right| 19.14 : 1
|-
|  Jack-high ||align=right|  210 ||align=right|  215,040 ||     8.27%   ||   18.2%    ||align=right| 11.09 : 1
|-
| Queen-high ||align=right|  330 ||align=right|  337,920 ||    13.0%    ||   31.2%    ||align=right| 6.69 : 1
|-
|  King-high ||align=right|  495 ||align=right|  506,880 ||    19.5%    ||   50.7%    ||align=right| 4.13 : 1
|-
|'''Total''' ||align=right|1,287 ||align=right|1,317,888 ||    50.7%    ||   50.7%    ||align=right| 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.

===7-card lowball poker hands===
{{See also|Lowball (poker)}}

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 <math display="inline">{52 \choose 7} = 133,784,560</math>, 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.<!-- I suggest another line for "pair or better" so that the table total is 100% -->

:{| class="wikitable" style="font-size: 95%;"
|-
!Hand        !!               Frequency !! Probability !! Cumulative !! Odds against
|-
|     5-high ||align=right|     781,824 ||     0.584%  ||    0.584%  ||align=right| 170.12 : 1
|-
|     6-high ||align=right|   3,151,360 ||     2.36%   ||    2.94%   ||align=right| 41.45 : 1
|-
|     7-high ||align=right|   7,426,560 ||     5.55%   ||    8.49%   ||align=right| 17.01 : 1
|-
|     8-high ||align=right|  13,171,200 ||     9.85%   ||   18.3%    ||align=right| 9.16 : 1
|-
|     9-high ||align=right|  19,174,400 ||    14.3%    ||   32.7%    ||align=right| 5.98 : 1
|-
|    10-high ||align=right|  23,675,904 ||    17.7%    ||   50.4%    ||align=right| 4.65 : 1
|-
|  Jack-high ||align=right|  24,837,120 ||    18.6%    ||   68.9%    ||align=right| 4.39 : 1
|-
| Queen-high ||align=right|  21,457,920 ||    16.0%    ||   85.0%    ||align=right| 5.23 : 1
|-
|  King-high ||align=right|  13,939,200 ||    10.4%    ||   95.4%    ||align=right| 8.60 : 1
|-
|'''Total''' ||align=right| 127,615,488 ||    95.4%    ||   95.4%    ||align=right| 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.

==See also==
{{Portal|Games}}
* [[Binomial coefficient]]
* [[Combination]]
* [[Combinatorial game theory]]
* [[Effective hand strength algorithm]]
* [[Event (probability theory)]]
* [[Game complexity]]
* [[Gaming mathematics]]
* [[Odds]]
* [[Permutation]]
* [[Probability]]
* [[Sample space]]
* [[Set theory]]

==References==
{{reflist}}

==External links==
* [https://www.math.sfu.ca/~alspach/ Brian Alspach's mathematics and poker page] {{Webarchive|url=https://web.archive.org/web/20150409230719/http://www.math.sfu.ca/~alspach/ |date=9 April 2015 }}
* [http://mathworld.wolfram.com/Poker.html MathWorld: Poker]
* [http://www.probabilityof.com/poker.shtml Poker probabilities including conditional calculations]
* [http://www.suffe.cool/poker/table1.html Numerous poker probability tables]
* [http://www.durangobill.com/Poker.html 5, 6, and 7 card poker probabilities]
* [http://probability.infarom.ro/holdempoker.html Hold'em poker probabilities]

{{Poker}}
{{Gambling}}
{{Use dmy dates|date=September 2019}}

{{DEFAULTSORT:Poker Probability}}
[[Category:Poker probability| ]]