Editing Poker probability (section)

===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.