Poker Hand Rankings Wikipedia
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*Poker Hand Rankings Wikipedia 2019
Non-standard poker hands are hands which are not recognized by official poker rules but are made by house rules. Non-standard hands usually appear in games using wild cards or bugs. Other terms for nonstandard hands are special hands or freak hands. Because the hands are defined by house rules, the composition and ranking of these hands is subject to variation. Any player participating in a game with non-standard hands should be sure to determine the exact rules of the game before play begins.Types[edit]
The usual hierarchy of poker hands from highest to lowest runs as follows (standard poker hands are in italics):
The is the best possible hand you can get in standard five-card Poker is called a royal flush. This hand consists of an: ace, king, queen, jack and 10, all of the same suit. If you have a royal flush, you’ll want to bet higher because this is a hard hand to beat. Liliboas / Getty Images. The usual hierarchy of poker hands from highest to lowest runs as follows (standard poker hands are in italics ): Royal Flush: See Straight Flush. Skeet flush: The same cards as a skeet (see below) but all in the same suit. Straight flush: The highest straight flush, A-K-Q-J-10 suited, is also.
*Royal Flush: SeeStraight Flush.
*Skeet flush: The same cards as a skeet (see below) but all in the same suit.
*Straight flush: The highest straight flush, A-K-Q-J-10 suited, is also called a royal flush. When wild cards are used, a wild card becomes whichever card is necessary to complete the straight flush, or the higher of the two cards that can complete an open-ended straight flush. For example, in the hand 10♠ 9♠ (Wild) 7♠ 6♠, it becomes the 8♠, and in the hand (Wild) Q♦ J♦ 10♦ 9♦, it plays as the K♦ (even though the 8♦ would also make a straight flush).
*Four of a kind: Between two equal sets of four of a kind (possible in wild card and community card poker games or with multiple or extended decks), the kicker determines the winner.
*Big bobtail: A four card straight flush (four cards of the same suit in consecutive order).
*Flush: When wild cards are used, a wild card contained in a flush is considered to be of the highest rank not already present in the hand. For example, in the hand (Wild) 10♥ 8♥ 5♥ 4♥, the wild card plays as the A♥, but in the hand A♣ K♣ (Wild) 9♣ 6♣, it plays as the Q♣. (As noted above, if a wild card would complete a straight flush, it will play as the card that would make the highest possible hand.) A variation is the double-ace flush rule, in which a wild card in a flush always plays as an ace, even if one is already present (unless the wild card would complete a straight flush). In such a game, the hand A♠ (Wild) 9♠ 5♠ 2♠ would defeat A♦ K♦ Q♦ 10♦ 8♦ (the wild card playing as an imaginary second A♠), whereas by the standard rules it would lose (because even with the wild card playing as a K♠, the latter hand’s Q♦ outranks the former’s 9♠).
*Straight Flush House: Same as Flush House (see below), but all cards are in consecutive order.
*Big cat: See cats and dogs below.
*Little cat: See cats and dogs below.
*Big dog: See cats and dogs below.
*Little dog: See cats and dogs below.
*Straight: When wild cards are used, the wild card becomes whichever rank is necessary to complete the straight. If two different ranks would complete a straight, it becomes the higher. For example, in the hand J♦ 10♠ 9♣ (Wild) 7♠, the wild card plays as an 8 (of any suit; it doesn’t matter). In the hand (Wild) 6♥ 5♦ 4♥ 3♦, it plays as a 7 (even though a 2 would also make a straight).
*Wrap-around straight: Also called a round-the-corner straight, consecutive cards including an ace which counts as both the high and low card. (Example Q-K-A-2-3).
*Skip straight: Also called alternate straight, Dutch straight, skipper, or kangaroo straight, Cards are in consecutive order, skipping every second rank (example 3-5-7-9-J).
*Five and dime: 5-low, 10-high, with no pair (example 5-6-7-8-10).[1]
*Skeet: Also called pelter or bracket, a hand with a deuce (2), a 5, and a 9, plus two other un-paired cards lower than 9 (example 2-4-5-6-9).[2]
*Little bobtail: A three card straight flush (three cards of the same suit in consecutive order).
*Flash: One card of each suit plus a joker.
*Blaze: Also called blazer, all cards are jacks, queens, and/or kings.
*Bobtail flush: Also called four flush, Four cards of the same suit.
*Flush house: Three cards of one suit and two cards of another.
*Bobtail straight: Also called four straight, four cards in consecutive order.
Some poker games are played with a deck that has been stripped of certain cards, usually low-ranking ones. For example, the Australian game of Manila uses a 32-card deck in which all cards below the rank of 7 are removed, and Mexican Stud removes the 8s, 9s, and 10s. In both of these games, a flush ranks above a full house, because having fewer cards of each suit available makes full houses more common.Cats and dogs[edit]
’Cats’ (or ’tigers’) and ’dogs’ are types of no-pair hands defined by their highest and lowest cards. The remaining three cards are kickers. Dogs and cats rank above straights and below Straight Flush houses. Usually, when cats and dogs are played, they are the only unconventional hands allowed.
*Little dog: Seven high, two low (for example, 7-6-4-3-2). It ranks just above a straight, and below a Straight Flush House or any other cat or dog. In standard poker seven high is the lowest hand possible.
*Big dog: Ace high, nine low (for example, A-K-J-10-9). Ranks above a straight or little dog, and below a Straight Flush House or cat.
*Little cat (or little tiger): Eight high, three low. Ranks above a straight or any dog, but below a Straight Flush House or big cat.
*Big cat (or big tiger): King high, eight low. It ranks just below a Straight Flush House, and above a straight or any other cat or dog.
Some play that dog or cat flushes beat a straight flush, under the reasoning that a plain dog or cat beats a plain straight. This makes the big cat flush the highest hand in the game.Kilters[edit]
A Kilter, also called Kelter, is a generic term for a number of different non-standard hands. Depending on house rules, a Kilter may be a Skeet, a Little Cat, a Skip Straight, or some variation of one of these hands.See also[edit]References[edit]
*^1897-1985, Gibson, Walter B. (Walter Brown) (2013-10-23). Hoyle’s modern encyclopedia of card games : rules of all the basic games and popular variations. ISBN978-0307486097. OCLC860901380.CS1 maint: numeric names: authors list (link)
*^Stevens, Michael (November 3, 2018). ’15 Poker Hand Names That Will Make You Smile (And Where Those Names Came From)’. gamblingsites.org. Retrieved February 19, 2019.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Non-standard_poker_hand&oldid=988247608’
This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities
Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.
___________________________________________________________________________Preliminary Calculation
Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.These are the same hand. Order is not important.
The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.
The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.
Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is
This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.
Imac oneida casino. The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.
If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.
Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.
Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:
One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.
___________________________________________________________________________The Poker Hands
Here’s a ranking chart of the Poker hands.
The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.
Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.
The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.
Definitions of Poker HandsPoker HandDefinition1Royal FlushA, K, Q, J, 10, all in the same suit2Straight FlushFive consecutive cards,all in the same suit3Four of a KindFour cards of the same rank,one card of another rank4Full HouseThree of a kind with a pair5FlushFive cards of the same suit,not in consecutive order6StraightFive consecutive cards,not of the same suit7Three of a KindThree cards of the same rank,2 cards of two other ranks8Two PairTwo cards of the same rank,two cards of another rank,one card of a third rank9One PairThree cards of the same rank,3 cards of three other ranks10High CardIf no one has any of the above hands,the player with the highest card wins
___________________________________________________________________________Counting Poker Hands
Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.
Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.
Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is
Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?
Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.
Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.
Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.
Two Pair and One Pair
These two are left as exercises.
High Card
The count is the complement that makes up 2,598,960.
The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.
Probabilities of Poker HandsPoker HandCountProbability2Straight Flush400.00001543Four of a Kind6240.00024014Full House3,7440.00144065Flush5,1080.00196546Straight10,2000.00392467Three of a Kind54,9120.02112858Two Pair123,5520.04753909One Pair1,098,2400.422569010High Card1,302,5400.5011774Total2,598,9601.0000000Poker Hand Rankings Wikipedia 2019
___________________________________________________________________________
2017 – Dan Ma
Register here: http://gg.gg/xaait
https://diarynote-jp.indered.space
*Poker Hand Rankings Wikipedia 2019
Non-standard poker hands are hands which are not recognized by official poker rules but are made by house rules. Non-standard hands usually appear in games using wild cards or bugs. Other terms for nonstandard hands are special hands or freak hands. Because the hands are defined by house rules, the composition and ranking of these hands is subject to variation. Any player participating in a game with non-standard hands should be sure to determine the exact rules of the game before play begins.Types[edit]
The usual hierarchy of poker hands from highest to lowest runs as follows (standard poker hands are in italics):
The is the best possible hand you can get in standard five-card Poker is called a royal flush. This hand consists of an: ace, king, queen, jack and 10, all of the same suit. If you have a royal flush, you’ll want to bet higher because this is a hard hand to beat. Liliboas / Getty Images. The usual hierarchy of poker hands from highest to lowest runs as follows (standard poker hands are in italics ): Royal Flush: See Straight Flush. Skeet flush: The same cards as a skeet (see below) but all in the same suit. Straight flush: The highest straight flush, A-K-Q-J-10 suited, is also.
*Royal Flush: SeeStraight Flush.
*Skeet flush: The same cards as a skeet (see below) but all in the same suit.
*Straight flush: The highest straight flush, A-K-Q-J-10 suited, is also called a royal flush. When wild cards are used, a wild card becomes whichever card is necessary to complete the straight flush, or the higher of the two cards that can complete an open-ended straight flush. For example, in the hand 10♠ 9♠ (Wild) 7♠ 6♠, it becomes the 8♠, and in the hand (Wild) Q♦ J♦ 10♦ 9♦, it plays as the K♦ (even though the 8♦ would also make a straight flush).
*Four of a kind: Between two equal sets of four of a kind (possible in wild card and community card poker games or with multiple or extended decks), the kicker determines the winner.
*Big bobtail: A four card straight flush (four cards of the same suit in consecutive order).
*Flush: When wild cards are used, a wild card contained in a flush is considered to be of the highest rank not already present in the hand. For example, in the hand (Wild) 10♥ 8♥ 5♥ 4♥, the wild card plays as the A♥, but in the hand A♣ K♣ (Wild) 9♣ 6♣, it plays as the Q♣. (As noted above, if a wild card would complete a straight flush, it will play as the card that would make the highest possible hand.) A variation is the double-ace flush rule, in which a wild card in a flush always plays as an ace, even if one is already present (unless the wild card would complete a straight flush). In such a game, the hand A♠ (Wild) 9♠ 5♠ 2♠ would defeat A♦ K♦ Q♦ 10♦ 8♦ (the wild card playing as an imaginary second A♠), whereas by the standard rules it would lose (because even with the wild card playing as a K♠, the latter hand’s Q♦ outranks the former’s 9♠).
*Straight Flush House: Same as Flush House (see below), but all cards are in consecutive order.
*Big cat: See cats and dogs below.
*Little cat: See cats and dogs below.
*Big dog: See cats and dogs below.
*Little dog: See cats and dogs below.
*Straight: When wild cards are used, the wild card becomes whichever rank is necessary to complete the straight. If two different ranks would complete a straight, it becomes the higher. For example, in the hand J♦ 10♠ 9♣ (Wild) 7♠, the wild card plays as an 8 (of any suit; it doesn’t matter). In the hand (Wild) 6♥ 5♦ 4♥ 3♦, it plays as a 7 (even though a 2 would also make a straight).
*Wrap-around straight: Also called a round-the-corner straight, consecutive cards including an ace which counts as both the high and low card. (Example Q-K-A-2-3).
*Skip straight: Also called alternate straight, Dutch straight, skipper, or kangaroo straight, Cards are in consecutive order, skipping every second rank (example 3-5-7-9-J).
*Five and dime: 5-low, 10-high, with no pair (example 5-6-7-8-10).[1]
*Skeet: Also called pelter or bracket, a hand with a deuce (2), a 5, and a 9, plus two other un-paired cards lower than 9 (example 2-4-5-6-9).[2]
*Little bobtail: A three card straight flush (three cards of the same suit in consecutive order).
*Flash: One card of each suit plus a joker.
*Blaze: Also called blazer, all cards are jacks, queens, and/or kings.
*Bobtail flush: Also called four flush, Four cards of the same suit.
*Flush house: Three cards of one suit and two cards of another.
*Bobtail straight: Also called four straight, four cards in consecutive order.
Some poker games are played with a deck that has been stripped of certain cards, usually low-ranking ones. For example, the Australian game of Manila uses a 32-card deck in which all cards below the rank of 7 are removed, and Mexican Stud removes the 8s, 9s, and 10s. In both of these games, a flush ranks above a full house, because having fewer cards of each suit available makes full houses more common.Cats and dogs[edit]
’Cats’ (or ’tigers’) and ’dogs’ are types of no-pair hands defined by their highest and lowest cards. The remaining three cards are kickers. Dogs and cats rank above straights and below Straight Flush houses. Usually, when cats and dogs are played, they are the only unconventional hands allowed.
*Little dog: Seven high, two low (for example, 7-6-4-3-2). It ranks just above a straight, and below a Straight Flush House or any other cat or dog. In standard poker seven high is the lowest hand possible.
*Big dog: Ace high, nine low (for example, A-K-J-10-9). Ranks above a straight or little dog, and below a Straight Flush House or cat.
*Little cat (or little tiger): Eight high, three low. Ranks above a straight or any dog, but below a Straight Flush House or big cat.
*Big cat (or big tiger): King high, eight low. It ranks just below a Straight Flush House, and above a straight or any other cat or dog.
Some play that dog or cat flushes beat a straight flush, under the reasoning that a plain dog or cat beats a plain straight. This makes the big cat flush the highest hand in the game.Kilters[edit]
A Kilter, also called Kelter, is a generic term for a number of different non-standard hands. Depending on house rules, a Kilter may be a Skeet, a Little Cat, a Skip Straight, or some variation of one of these hands.See also[edit]References[edit]
*^1897-1985, Gibson, Walter B. (Walter Brown) (2013-10-23). Hoyle’s modern encyclopedia of card games : rules of all the basic games and popular variations. ISBN978-0307486097. OCLC860901380.CS1 maint: numeric names: authors list (link)
*^Stevens, Michael (November 3, 2018). ’15 Poker Hand Names That Will Make You Smile (And Where Those Names Came From)’. gamblingsites.org. Retrieved February 19, 2019.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Non-standard_poker_hand&oldid=988247608’
This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities
Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.
___________________________________________________________________________Preliminary Calculation
Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.These are the same hand. Order is not important.
The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.
The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.
Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is
This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.
Imac oneida casino. The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.
If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.
Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.
Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:
One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.
___________________________________________________________________________The Poker Hands
Here’s a ranking chart of the Poker hands.
The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.
Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.
The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.
Definitions of Poker HandsPoker HandDefinition1Royal FlushA, K, Q, J, 10, all in the same suit2Straight FlushFive consecutive cards,all in the same suit3Four of a KindFour cards of the same rank,one card of another rank4Full HouseThree of a kind with a pair5FlushFive cards of the same suit,not in consecutive order6StraightFive consecutive cards,not of the same suit7Three of a KindThree cards of the same rank,2 cards of two other ranks8Two PairTwo cards of the same rank,two cards of another rank,one card of a third rank9One PairThree cards of the same rank,3 cards of three other ranks10High CardIf no one has any of the above hands,the player with the highest card wins
___________________________________________________________________________Counting Poker Hands
Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.
Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.
Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is
Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?
Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.
Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.
Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.
Two Pair and One Pair
These two are left as exercises.
High Card
The count is the complement that makes up 2,598,960.
The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.
Probabilities of Poker HandsPoker HandCountProbability2Straight Flush400.00001543Four of a Kind6240.00024014Full House3,7440.00144065Flush5,1080.00196546Straight10,2000.00392467Three of a Kind54,9120.02112858Two Pair123,5520.04753909One Pair1,098,2400.422569010High Card1,302,5400.5011774Total2,598,9601.0000000Poker Hand Rankings Wikipedia 2019
___________________________________________________________________________
2017 – Dan Ma
Register here: http://gg.gg/xaait
https://diarynote-jp.indered.space
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