Hand (poker)

In the card game of poker , the term hand describes the best five cards a player can use. The ranking of the individual card combinations is the same for all game variants, only their probability varies. The most important change is a deck with a wild card . The best possible hand that can be reached in such a deck is five of a kind, while the best hand in a normal French hand is a royal flush.

General rules

• If there is a tie in the rank of the hand, the size of the individual cards usually decides. The following order in descending order applies: Ace - King - Queen - Jack - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 . If such a card is decisive, the affected player has the better kicker .
• A hand always consists of five cards. Therefore it is not possible for the kicker to decide on two equally high straights , as these already consist of five cards.
• Cards are first scored according to the rank of the hand and only then according to the number of cards involved: Two pairs of twos and threes are therefore better than a pair of aces.
• There is no hierarchy of symbols that affects the rank of the hand, the strength of a flush does not depend on which of the four symbols the five cards of the same color belong to.

Influence of the game variants on the probabilities of the hands

The information on the probabilities of the different hands depends on the game variant; are therefore dependent on whether there are community cards (e.g. Texas Hold'em ) or whether cards can be exchanged (e.g. Draw Poker ).

The different game variants are characterized by the fact that there are different ways to get to a hand of five cards. The total number of combinations therefore changes from game variant to game variant. If you just draw five cards from a poker hand of 52 cards, there are 2,598,960 combinations, with seven cards from 52 (Texas Hold'em) there are already 133,784,560 combinations.

In general, if you divide the number of combinations for a hand by the total number of combinations, this gives the probability of receiving this hand in this variant of the game. So the rule (with exceptions) usually applies: every hand is more valuable the fewer combinations it corresponds to.

Both the ability to swap cards and the option to choose from community cards usually tend to influence the odds in favor of more valuable hands. So in the game variant Texas Hold'em z. B. the pair more likely than high card , but still ranks higher, since mind you, the ranking of the individual hands always remains unchanged.

In the case of the card exchange, it naturally depends on the strategy chosen by the player how the probabilities are influenced in detail. A calculation that is valid independently of the player's strategy is therefore not possible, and the determination of a possibly optimal exchange strategy cannot be discussed here.

In the case of community cards , too, the probabilities are a lot more complicated to calculate than in the case of 5 out of 52. For 7 out of 52 ( Texas Hold'em ), the two pair and high card hands are compared here only as an example .

Possible combinations with five out of 52 cards

If you draw five cards from a poker hand of 52 cards, 2,598,960 combinations are possible:

${\ displaystyle {52 \ choose 5} = {{52!} \ over {5! (52-5)!}} = {\ frac {52 \ cdot 51 \ cdot 50 \ cdot 49 \ cdot 48} {5 \ cdot 4 \ cdot 3 \ cdot 2 \ cdot 1}} = 2,598,960}$

The following table shows the number of possibilities for each hand to be formed with 5 of 52 cards; with other variants (for example adding jokers or shuffling several complete decks of cards) other values ​​would result. The next column shows the probability resulting from this number of receiving such a hand by randomly drawing five cards; Variants with strategic behavior or choices are not considered here. For these restrictions, see the section Influence of the game variants on the probabilities . The next column “as a ratio” gives the probability of such a hand not as a percentage, but in the form of odds . Finally, the cumulative probability indicates how likely it is to draw at least the combination under consideration. The table counts the extremely rare royal flush in the straight flush, which is justified in that it is the highest of the straight flushes even if it is not specifically named.

hand Number of
possibilities
probability cumulative probability
in percent as an inverted odd in percent as an inverted odd
Straight Flush & Royal Flush 36 + 4 000.00154% 64,973: 1 000.00154% 64,973: 1
Quadruplets 624 000.0240% 4,164: 1 000.0255% 3,913: 1
Full house 3,744 000.144% 693.2: 1 000.170% 588.6: 1
Flush 5,108 000.197% 507.8: 1 000.366% 272.1: 1
Straight 10,200 000.392% 253.8: 1 000.76% 130.8: 1
Triplet 54,912 002.11% 46.3: 1 002.87% 33.8: 1
Two couples 123,552 004.75% 20.0: 1 007.63% 12.1: 1
A few 1,098,240 042.3% 1.37: 1 049.9% 1.005: 1
Supreme card 1,302,540 050.1% 0.995: 1 100% 0: 1
total 2,598,960 100% 0: 1 - -

Royal flush

This hand is actually a straight flush, but it is considered separately because of its role as the best hand in poker and its rarity. A royal flush , such as B. A ♣ K ♣ Q ♣ J ♣ 10 ♣, is a straight flush with the ace as the highest card, thus the highest straight flush.

In the very unlikely event that e.g. B. In Draw Poker , if two players hold a royal flush, the pot is split. In the Hold'em variants, in which community cards are played, such a situation is only possible if the royal flush is completely open on the table, i.e. the five community cards ( board ) show the royal flush, in the Omaha Hold variant 'em , in which there are also community cards, such a situation is not possible.

Examples:

• A ♣ K ♣ Q ♣ J ♣ 10 ♣ beats K ♣ Q ♣ J ♣ 10 ♣ 9 ♣

A splitpot is only possible if the board is A ♣ K ♣ Q ♣ J ♣ 10 ♣ (or other color). In this case, all players play the royal flush from the board.

Number of possible combinations

There is a possible highest card (ace) and four different suits:

${\ displaystyle {1 \ choose 1} {4 \ choose 1} = 4}$

Straight flush

The various straight flushes (for: single-colored streets; including the royal flush , see above) are the best possible card combinations. An example is a hand like Q ♠ J ♠ 10 ♠ 9 ♠ 8 ♠ that contains five cards in a row of the same suit. Two competing straight flushes are rated according to their highest card, similar to a straight . The probability of a straight flush occurring is even less than that of four cards of the same rank (e.g. four jacks), so the straight flush is the second highest rated of all poker hands. Straights with 5 as the highest card are also possible here, such as 5 ♦ 4 ♦ 3 ♦ 2 ♦ A ♦ . This hand is also known as the steel wheel .

Examples:

• 7 ♥ 6 ♥ 5 ♥ 4 ♥ 3 ♥ beats 5 ♠ 4 ♠ 3 ♠ 2 ♠ A ♠
• J ♣ 10 ♣ 9 ♣ 8 ♣ 7 ♣ "splits" J ♦ 10 ♦ 9 ♦ 8 ♦ 7 ♦ ( split pot )

Number of possible combinations (without royal flush)

There are nine different possible highest cards and four different suits (without ace as the highest card) :

${\ displaystyle {9 \ choose 1} {4 \ choose 1} = 36}$

A Vierling , or poker , in English and four of a kind or quads called, is another poker hand. An example of this is 9 ♣ 9 ♠  9 ♦ 9 ♥ J ♥ . Four of a kind contains four cards of the same value. The four of a kind is above the full house and below a straight flush . It decides the amount of the quadruplet. If there is already four of a kind under the community cards so that all remaining players can use this four of a kind, the height of the kicker decides; if they are tied, a split pot occurs .

Examples:

• 10 ♣  10 ♦ 10 ♥  10 ♠  5 ♦ beats 6 ♦ 6 ♥  6 ♠ 6 ♣ K ♠
• 10 ♣  10 ♦ 10 ♥  10 ♠ Q ♣ beats 10 ♣  10 ♦ 10 ♥  10 ♠  5 ♦ because of the better kicker
• 10 ♣  10 ♦ 10 ♥  10 ♠ Q ♣ "splits" 10 ♣  10 ♦ 10 ♥  10 ♠  Q ♦ ( split pot )

Number of possible combinations

Each of the thirteen values ​​can develop into four of a kind. Stay (52-4) = 48 remaining cards that serve as kicker:

${\ displaystyle {13 \ choose 1} {48 \ choose 1} = 624}$

Another approach - equivalent to twins and triplets - assumes that each of the thirteen values ​​can form a four of a kind. It contains four of the four colors of a value. The remaining card can have one of the twelve remaining values ​​in four different colors:

${\ displaystyle {13 \ choose 1} {4 \ choose 4} {12 \ choose 1} {4 \ choose 1} = 624}$

Full house

A full house of German full house , sometimes full boat called, corresponds to a hand such as 3 ♣ 3 ♠ 3 ♦ 6 ♣ 6 ♥ . So a full house consists of three of a kind and a pair . This means that the hand is valued below a four of a kind and above a flush . The amount of triplets decides. If two players can use the community cards to put together a full house with the same three of a kind, the amount of the pair decides; in the event of a tie there is a split pot .

Examples:

• 10 ♠ 10 ♥ 10 ♦ 4 ♠ 4 ♦ beats 9 ♥ 9 ♣ 9 ♠ A ♥ A ♣ because of the better triplet
• 10 ♠ 10 ♥ 10 ♦ 4 ♠ 4 ♦ beats 10 ♥ 10 ♣ 10 ♠ 3 ♥ 3 ♣ because of the better pair
• Q ♥ Q ♦ Q ♣ 8 ♥ 8 ♣ "splits" Q ♥ Q ♦ Q ♠ 8 ♥ 8 ♣ ( split pot )

Number of possible combinations

The triplet can be of thirteen values ​​and three different colors. The pair can be one of the remaining twelve values ​​and consists of two of four colors:

${\ displaystyle {13 \ choose 1} {4 \ choose 3} {12 \ choose 1} {4 \ choose 2} = 3.744}$

Flush

A flush is a hand such as Q ♣ 10 ♣ 7 ♣ 6 ♣ 4 ♣ that is made up of five cards of the same suit. Two flushes are ranked on their highest card. If this is the same, the second highest card decides, then the third highest card and so on. A flush does not have to be made from consecutive cards. But if that is the case, it is called a straight flush . The color of the flush does not matter in the order.

Examples:

• A ♥ Q ♥ 10 ♥ 5 ♥ 3 ♥ beats K ♠ Q ♠ J ♠ 9 ♠ 6 ♠ ( ace high flush wins)
• A ♦ K ♦ 7 ♦ 6 ♦ 2 ♦ beats A ♥ Q ♥ 10 ♥ 5 ♥ 3 ♥ ( flush, ace king high wins)
• Q ♥ 10 ♥ 9 ♥ 5 ♥ 2 ♥ "splits" Q ♠ 10 ♠ 9 ♠ 5 ♠ 2 ♠ ( split pot )

Number of possible combinations

The flush consists of five cards of the same suit. There are thirteen cards of each suit. There are four different colors. From the number we subtract the 36 possible straight flushes and the four possible royal flushes , each of which is evaluated separately:

${\ displaystyle {13 \ choose 5} {4 \ choose 1} -36-4 = 5,108}$

Straight

A Straight , in Germans road is a hand such as Q ♣ J ♠ 10 ♠ 9 ♥ 8 ♥ that various of five consecutive map colors is formed. However, if the suits of the five cards are identical, it is called a straight flush . The hand is stronger than three of a kind and weaker than a flush . If there are two straights in circulation, the highest card is used. If this is the same, there is a split pot . Straights with five as the highest number, i.e. A - 2 - 3 - 4 - 5, are permitted, but straights such as K - A - 2 - 3 - 4 ( round the corner straight ) are not permitted unless expressly agreed. Other variants, such as skip straight (3 - 5 - 7 - 9 - J) should also be agreed before the start of the game round, if necessary including their evaluation.

Examples:

• 8 ♠ 7 ♠ 6 ♥ 5 ♥ 4 ♠ beats 6 ♦ 5 ♠ 4 ♦ 3 ♥ 2 ♣ ( eight high straight )
• 8 ♠ 7 ♠ 6 ♥ 5 ♥ 4 ♠ “split” 8 ♥ 7 ♦ 6 ♣ 5 ♣ 4 ♥ ( split pot )

Number of possible combinations

A straight consists of five cards. It consists of one of the ten possible highest cards. Each card can be any of the four colors. As with the flushes , the 36 straight flushes and the four royal flushes are deducted:

${\ displaystyle {10 \ choose 1} {4 \ choose 1} ^ {5} -36-4 = 10,200}$

Triplet

Drilling also in English three of a kind or trips called a hand as 2 ♦ 2 ♠ 2 ♥ K ♠ 6 ♠ containing three cards of the same value and two other cards. It is placed above the two pairs and below the straight . If two players can form three of a kind from the community cards, the height of the first kicker decides, in the event of a tie, the second kicker.

Examples:

• 8 ♠ 8 ♥ 8 ♦ 5 ♠ 3 ♣ beats 5 ♣ 5 ♥ 5 ♦ Q ♦ 10 ♣ ( three eights wins)
• 8 ♠ 8 ♥ 8 ♦ A ♣ 2 ♦ takes 8 ♣ 8 ♥ 8 ♦ 5 ♠ 3 ♣

Although identical in terms of valuation, games with community cards result in two fundamentally different game situations.

• A set is three of a kind created from a pocket pair , which is a very strong hand, especially since it is difficult to read for the opponent.
• Trips is three of a kind with a starting hand card and an open pair. This combination can never be the nuts .

Number of possible combinations

Any of the thirteen values ​​can form a triplet. It contains three of the four colors of a value. The other two cards must have two of the twelve remaining values ​​and can be of four different suits:

${\ displaystyle {13 \ choose 1} {4 \ choose 3} {12 \ choose 2} {4 \ choose 1} ^ {2} = 54,912}$

Two couples

A hand like J ♥ J ♣ 4 ♣ 4 ♠ 9 ♠ is called two pairs . two pair . Often times the pairs are also mentioned, such as two pairs, aces and eights . It consists of two pairs and one other card. If there are several double pairs, the higher pair decides, then the second highest and, if necessary, the kicker. The hand is placed below the three of a kind and above the pair .

Examples:

• K ♥ K ♦ 2 ♣ 2 ♦ J ♥ beats J ♦ J ♠ 10 ♠ 10 ♣ 9 ♠ ( kings up wins)
• 4 ♠ 4 ♣ 3 ♠ 3 ♥ K ♦ ( fours and threes, king kicker ) beats 4 ♥ 4 ♦ 3 ♦ 3 ♣ 10 ♠

Number of possible combinations

Each of the two pairs can have one of the thirteen values ​​and two of the four colors. The kicker can have one of the eleven remaining values ​​and any color:

${\ displaystyle {13 \ choose 2} {4 \ choose 2} ^ {2} {11 \ choose 1} {4 \ choose 1} = 123,552}$

A few

A couple , Eng. one pair , is a hand that has a double value, such as 4 ♥ 4 ♠ K ♠ 10 ♦ 5 ♠ that also contains three other cards. The hand is weaker than two pair and better than the so-called high card . If two players can present pairs of the same number, the height of the first kicker decides, in the event of a tie the second and possibly the third kicker.

Examples:

• 10 ♣ 10 ♠ 6 ♠ 4 ♥ 2 ♥ beats 9 ♥ 9 ♣ A ♥ Q ♦ 10 ♦
• 10 ♥ 10 ♦ J ♦ 3 ♥ 2 ♣ beats 10 ♣ 10 ♠ 6 ♠ 4 ♥ 2 ♥
• 10 ♥ 10 ♦ J ♦ 4 ♥ 3 ♣ beats 10 ♣ 10 ♠ J ♠ 4 ♦ 2 ♥

Number of possible combinations

A pair can have thirteen values ​​and two of four different colors. The remaining three cards can have twelve different values ​​and four suits:

${\ displaystyle {13 \ choose 1} {4 \ choose 2} {12 \ choose 3} {4 \ choose 1} ^ {3} = 1,098,240}$

High card

A high card , also called no pair , does not mean any of the above combinations. An example is K ♥ J ♣ 8 ♣ 7 ♦ 3 ♠. If there are two competing high cards, the kicker counts, if there is a tie the second kicker and so on.

Examples:

• A ♦ 10 ♦ 9 ♠ 5 ♣ 4 ♣ beats K ♣ Q ♦ J ♣ 8 ♥ 7 ♥
• A ♦ 10 ♦ 9 ♠ 5 ♣ 4 ♣ beats A ♣ 9 ♦ 8 ♥ 5 ♠ 4 ♠

Number of possible combinations

The number results from the difference between the number of all possible hands and the sum of the hands listed above from royal flush to pair:

${\ displaystyle {52 \ choose 5} -1,296,420 = 1,302,540}$

Another approach is to look at values ​​and colors independently of one another:

• There must be five different values, but none of the ten streets - if you only look at the values ​​and leave out the colors .
• The colors must not form any of the four flushes, whereby here only the colors are considered and the values ​​are left out.

These two numbers are multiplied together:

${\ displaystyle \ left [{13 \ choose 5} -10 \ right] (4 ^ {5} -4) = 1,302,540}$

Possible combinations with 7 out of 52 cards (Texas Hold'em)

hand Number of
possibilities
probability cumulative
in percent as a ratio in percent
Straight Flush & Royal Flush 41,584 000.0311% 3216.2: 1 000.0311%
Quadruplets 224,848 000.168% 594: 1 000.199%
Full house 3,473,184 002.60% 37.5: 1 002.80%
Flush 4,047,644 003.03% 32.1: 1 005.82%
Straight 6.180.020 004.62% 20.6: 1 0010.44%
Triplet 6,461,620 004.83% 19.7: 1 0015.3%
Two couples 31,433,400 023.5% 3.3: 1 0038.8%
A few 58,627,800 043.8% 1.3: 1 0082.6%
Supreme card 23,294,460 017.4% 4.7: 1 00100%
total 133.784.560 100% 0: 1 -

Royal flush

There are four ways to form a royal flush from five cards. With the remaining two cards there are now options for any royal flush . So overall there is ${\ displaystyle {47 \ choose 2}}$

${\ displaystyle 4 \ cdot {47 \ choose 2} = 4324}$ possible combinations.

Since there are a total of different (poker) combinations, the probability is then approximately 0.00323%. ${\ displaystyle {52 \ choose 7} = 133784560}$

Straight flush

With five cards there are 9 options (no royal flush) with four suits each for a straight flush. The remaining cards are split among the remaining 46 cards (the next higher card of the same suit would form a higher straight flush). There is then

${\ displaystyle 9 \ times 4 \ times {46 \ choose 2} = 37260}$ different options for a straight flush.

In order for a four of a kind to be formed from seven cards, four identical values ​​must occur, a straight flush is therefore not to be considered.

There are 13 different quadruplets. Since four cards have already been assigned, there are still three cards that can be freely combined from the remaining 48 cards. This gives

${\ displaystyle {48 \ choose 3} \ cdot 13 = 224,848}$

different combinations.

If you divide by the total number of possibilities, the probability of being able to form four of a kind as the best hand in Texas Hold'em is approx. 0.168%. ${\ displaystyle {52 \ choose 7} = 133.784.560}$

Full house

There are three different ways to form a full house:

Three of a kind, a pair and two kickers

First there are 13 possibilities for the height of the triplet, of which four different are possible. Then there are 12 possibilities for the height of the pair, of which six different are possible. There are still options for distributing the kicker . So there is${\ displaystyle {11 \ choose 2} \ cdot 4 ^ {2}}$
${\ displaystyle 13 \ cdot {4 \ choose 3} \ cdot 12 \ cdot {4 \ choose 2} \ cdot {11 \ choose 2} \ cdot 4 ^ {2} = 3294720}$ Combinations.

Three of a kind and two pairs

There are again 13 possibilities for the height of the triplet. The two pairs are then distributed to the remaining 12 ranks. There are
${\ displaystyle 13 \ cdot {4 \ choose 3} \ cdot {12 \ choose 2} \ cdot {4 \ choose 2} ^ {2} = 123552}$
Combinations.

Two triplets and a kicker

The two triplets are first distributed to the 13 different ranks of the cards. The kicker can then still be one of the 44 remaining cards. There are
${\ displaystyle {13 \ choose 2} \ cdot {4 \ choose 3} ^ {2} \ cdot 11 \ cdot 4 = 54912}$
Combinations.

So overall there are possible combinations for a full house. ${\ displaystyle 3294720 + 123552 + 54912 = 3473184}$

That gives a probability of about 2.60%.

Flush

There are three options for a flush: exactly five cards of the same color, exactly six cards of the same color, exactly seven cards of the same color. The five, six or seven cards of the flush are first distributed to the 13 different rank levels of the cards. Then the number of combinations that would make a straight flush, creating a higher ranking hand, is subtracted. Now there are four different colors in which the flush can be. The remaining cards are now distributed to the 39 cards that would not form a flush with more cards of the same color. You get

${\ displaystyle \ left [{13 \ choose 5} -10 \ right] \ cdot 4 \ cdot {39 \ choose 2} + \ left [{13 \ choose 6} -9-8 \ cdot 6-2 \ cdot 7 \ right] \ cdot 4 \ cdot 39+ \ left [{13 \ choose 7} -8-7 \ cdot 5-2 \ cdot 6-8 \ cdot {6 \ choose 2} -2 \ cdot {7 \ choose 2 } \ right] \ cdot 4 = 4047644}$
Combinations for a flush.

Straight

There are 10 streets of different heights. A road can be built in three different ways:

One street and two kickers

In nine of the ten streets, no kicker is allowed to form a higher street. This is impossible on the highest street. The kickers may initially be all other cards, although no pair or triplet may be created together with a card of the original street. In addition, no flush may form. There are
${\ displaystyle 9 \ cdot (4 ^ {5} -4) \ cdot {28 \ choose 2} + (4 ^ {5} -4) \ cdot {32 \ choose 2} -9 \ cdot 4 \ cdot \ left [5 \ times 3 \ times \ left (7 \ times 3 \ times 7+ {7 \ choose 2} \ right) + {5 \ times choose 3} \ times 3 ^ {2} \ times {7 \ times choose 2} \ right] -4 \ times \ left [5 \ times 3 \ times \ left (8 \ times 3 \ times 8+ {8 \ choose 2} \ right) + {5 \ times choose 3} \ times 3 ^ {2} \ cdot {8 \ choose 2} \ right]}$
${\ displaystyle = 3793920}$ Combinations.

One of the two remaining cards forms a pair with a card of the street

There are five ways where the pair can be (rank level) and six ways each of how it can be divided among the colors. Taking into account that no flush may form again, there is
${\ displaystyle 9 \ cdot 5 \ cdot {4 \ choose 2} \ cdot \ left [(4 ^ {4} -2) \ cdot 28- \ left (2 \ cdot {4 \ choose 3} \ cdot 3 + 2 \ right) \ cdot 7 \ right] +5 \ cdot {4 \ choose 2} \ cdot \ left [(4 ^ {4} -2) \ cdot 32- \ left (2 \ cdot {4 \ choose 3} \ cdot 3 + 2 \ right) \ cdot 8 \ right] = 2108700}$
Combinations.

Both other cards form two pairs or three of a kind with one or two cards of the straight

There are ways where the two couples can be. In 24 out of 36 cases, one card from each pair is of the same suit. In six cases it is both. This must be considered in order to calculate the number of combinations that will make a flush. There are${\ displaystyle {5 \ choose 2}}$
${\ displaystyle 10 \ cdot {5 \ choose 2} \ cdot \ left [{4 \ choose 2} ^ {2} \ cdot 4 ^ {3} -24-6 \ cdot 2 \ right] +10 \ cdot 5 \ cdot {4 \ choose 3} \ cdot (4 ^ {4} -3) = 277400}$
Combinations.

So overall there are possible combinations for the straight poker hand. ${\ displaystyle 3793920 + 2108700 + 277400 = 6180020}$

That gives a probability of about 4.62%.

Triplet

Apart from three of a kind, there may be no other pairs, triplets or four of a kind, as this would result in higher-ranking combinations. The three of a kind and the four kickers are first distributed to the 13 different rankings of the cards. Then the number of combinations that would form a road must be subtracted. There are also five possibilities of where the three of a kind and the kickers can be (according to the rank of the cards). There are four possibilities for three of a kind (colors) with the same rank. There are still options for the kickers , although three must be subtracted because they would form a flush. There is then ${\ displaystyle 4 ^ {4}}$

${\ displaystyle \ left [{13 \ choose 5} -10 \ right] \ cdot {5 \ choose 1} \ cdot {4 \ choose 3} \ cdot (4 ^ {4} -3) = 6461620}$
Combinations for three of a kind.

Two couples

There are two ways that two pairs can be formed: two pairs and three kicker, three pairs and one kicker. In the first case, the two pairs and the three kickers are first distributed among the 13 rankings of the cards. The number of combinations that make up a street is subtracted. There are possibilities of where the pairs and the kickers are (rank height). Then the kicker is multiplied by the number of possible combinations (no flush may be formed). In the second case, the three pairs are first redistributed among the 13 ranks. There are six options (colors) for each pair. The kicker can now fall on 40 different cards. There are ${\ displaystyle {5 \ choose 2}}$

${\ displaystyle \ left [{13 \ choose 5} -10 \ right] \ cdot {5 \ choose 2} \ cdot \ left (6 \ cdot 4 ^ {3} +24 \ cdot (4 ^ {3} -1 ) +6 \ cdot (4 ^ {3} -2) \ right) + {13 \ choose 3} \ cdot 6 ^ {3} \ cdot 10 \ cdot 4 = 31433400}$
Combinations for two couples.

A few

So that a pair (but nothing better) can be formed from seven cards, six different values ​​must appear, one of which is double, but no straight. Ten streets in combination with another value with 13-5 possibilities are prohibited; here, however, nine “six-way streets” are deducted twice.

This gives for the values

${\ displaystyle \ left [{13 \ choose 6} -10 \ cdot (13-5) +9 \ right] \ cdot {6 \ choose 1} = 9870}$

different combinations.

With regard to the colors, you have to choose two of four colors for the cards that make up the pair , while everything is allowed for the other cards except four of the same colors with one of the two colors also appearing in the pair or otherwise five of the same colors.

This gives

${\ displaystyle {4 \ choose 2} \ cdot \ left [4 ^ {5} -2 \ cdot {5 \ choose 1} \ cdot 3-4 \ right] = 5940}$

different color combinations.

The total combinations result as a product, that is

${\ displaystyle 9870 \ cdot 5940 = 58,627,800}$

Combinations.

If you divide by the total number of options, there is a probability of approx. 43.8 percent of being able to form a pair as the best hand in Texas Hold'em. ${\ displaystyle {52 \ choose 7} = 133.784.560}$

High card

So that no valuable combination can be formed from seven cards, seven different values ​​must appear, but no street among them. According to the principle of inclusion and exclusion, you have to subtract the combinations of ten streets with two other values, then add the double-deducted combinations of nine six-streets with another value again. The eight streets of sevens are subtracted three times from the streets of five and added twice to the streets of six, so they have already been correctly deducted.

This gives

${\ displaystyle {13 \ choose 7} -10 \ cdot {8 \ choose 2} +9 \ cdot {7 \ choose 1} = 1499}$

Value combinations.

To stand

${\ displaystyle 4 ^ {7} -4 \ cdot {7 \ choose 5} \ cdot 3 ^ {2} -4 \ cdot {7 \ choose 6} \ cdot 3 ^ {1} -4 \ cdot {7 \ choose 7} \ cdot 3 ^ {0} = 15,540}$

Color combinations opposite.

Again as a product result

${\ displaystyle 1499 \ cdot 15,540 = 23,294,460}$

Combinations in total and thus a probability of only approx. 17.4 percent of only getting high card in Texas Hold'em.