Editing Poker probability (section)

===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).