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Interestingly, most, if not all, standard poker games like Texas Holdem make use of a single 52-card deck so it would be possible to compute for your exact probability of getting each hand. But we won’t bore you with the math behind it. You can learn all about that here.Instead, we’ll give you a quick summary of everything you need to know about poker hand odds. In our poker math and probability lesson it was stated that when it comes to poker; “the math is essential“. Although you don’t need to be a math genius to play poker, a solid understanding of probability will serve you well and knowing the odds is what it’s all about in poker.

• Probability Of A Pair In Poker
• List Of Poker Winning Hands
• Probability Of Winning Poker Hands Free
• Poker Hand Probabilities Explained

Contrary to what some poker strategists tend to preach you don’t need to memorize lists of odds and perform complex mathematics to be a winning Hold’em player. However, there are some simple Texas Hold’em odds and probabilities that you should know well when you’re drawing to a hand or want to prevent your opponents from doing so. A colleague of mine was recently playing at an online casino playing 10-line Jacks or Better Video Poker. Money was deposited and 10 hands were played. All 10 hands (and thus all 100 lines) failed to bring up a single win. Please can you calculate the probability of drawing a blank on 10 hands of 10 line JoB.

This page describes the ranking of poker hands. This applies not only in the game of poker itself, but also in certain other card games such as Chinese Poker, Chicago, Poker Menteur and Pai Gow Poker.

• Low Poker Ranking: A-5, 2-7, A-6
• Hand probabilities and multiple decks - probability tables

Standard Poker Hand Ranking

There are 52 cards in the pack, and the ranking of the individual cards, from high to low, is ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. In standard poker - that is to say in the formal casino and tournament game played internationally and the home game as normally played in North America - there is no ranking between the suits for the purpose of comparing hands - so for example the king of hearts and the king of spades are equal. (Note however that suit ranking is sometimes used for other purposes such as allocating seats, deciding who bets first, and allocating the odd chip when splitting a pot that can't be equally divided. See ranking of suits for details.)

A poker hand consists of five cards. The categories of hand, from highest to lowest, are listed below. Any hand in a higher category beats any hand in a lower category (so for example any three of a kind beats any two pairs). Between hands in the same category the rank of the individual cards decides which is better, as described in more detail below.

In games where a player has more than five cards and selects five to form a poker hand, the remaining cards do not play any part in the ranking. Poker ranks are always based on five cards only, and if these cards are equal the hands are equal, irrespective of the ranks of any unused cards.

Some readers may wonder why one would ever need to compare (say) two threes of a kind of equal rank. This obviously cannot arise in basic draw poker, but such comparisons are needed in poker games using shared (community) cards, such as Texas Hold'em, in poker games with wild cards, and in other card games using poker combinations.

1. Straight Flush

If there are no wild cards, this is the highest type of poker hand: five cards of the same suit in sequence - such as J-10-9-8-7. Between two straight flushes, the one containing the higher top card is higher. An ace can be counted as low, so 5-4-3-2-A is a straight flush, but its top card is the five, not the ace, so it is the lowest type of straight flush. The highest type of straight flush, A-K-Q-J-10 of a suit, is known as a Royal Flush. The cards in a straight flush cannot 'turn the corner': 4-3-2-A-K is not valid.

2. Four of a kind

Four cards of the same rank - such as four queens. The fifth card, known as the kicker, can be anything. This combination is sometimes known as 'quads', and in some parts of Europe it is called a 'poker', though this term for it is unknown in English. Between two fours of a kind, the one with the higher set of four cards is higher - so 3-3-3-3-A is beaten by 4-4-4-4-2. If two or more players have four of a kind of the same rank, the rank of the kicker decides. For example in Texas Hold'em with J-J-J-J-9 on the table (available to all players), a player holding K-7 beats a player holding Q-10 since the king beats the queen. If one player holds 8-2 and another holds 6-5 they split the pot, since the 9 kicker makes the best hand for both of them. If one player holds A-2 and another holds A-K they also split the pot because both have an ace kicker.

3. Full House

This combination, sometimes known as a boat, consists of three cards of one rank and two cards of another rank - for example three sevens and two tens (colloquially known as 'sevens full of tens' or 'sevens on tens'). When comparing full houses, the rank of the three cards determines which is higher. For example 9-9-9-4-4 beats 8-8-8-A-A. If the threes of a kind are equal, the rank of the pairs decides.

4. Flush

Five cards of the same suit. When comparing two flushes, the highest card determines which is higher. If the highest cards are equal then the second highest card is compared; if those are equal too, then the third highest card, and so on. For example K-J-9-3-2 beats K-J-7-6-5 because the nine beats the seven.If all five cards are equal, the flushes are equal.

5. Straight

Five cards of mixed suits in sequence - for example Q-J-10-9-8. When comparing two sequences, the one with the higher ranking top card is better. Ace can count high or low in a straight, but not both at once, so A-K-Q-J-10 and 5-4-3-2-A are valid straights, but 2-A-K-Q-J is not. 5-4-3-2-A, known as a wheel, is the lowest kind of straight, the top card being the five.

6. Three of a Kind

Three cards of the same rank plus two unequal cards. This combination is also known as Triplets or Trips. When comparing two threes of a kind the rank of the three equal cards determines which is higher. If the sets of three are of equal rank, then the higher of the two remaining cards in each hand are compared, and if those are equal, the lower odd card is compared.So for example 5-5-5-3-2 beats 4-4-4-K-5, which beats 4-4-4-Q-9, which beats 4-4-4-Q-8.

7. Two Pairs

A pair consists of two cards of equal rank. In a hand with two pairs, the two pairs are of different ranks (otherwise you would have four of a kind), and there is an odd card to make the hand up to five cards. When comparing hands with two pairs, the hand with the highest pair wins, irrespective of the rank of the other cards - so J-J-2-2-4 beats 10-10-9-9-8 because the jacks beat the tens. If the higher pairs are equal, the lower pairs are compared, so that for example 8-8-6-6-3 beats 8-8-5-5-K. Finally, if both pairs are the same, the odd cards are compared, so Q-Q-5-5-8 beats Q-Q-5-5-4.

8. Pair

A hand with two cards of equal rank and three cards which are different from these and from each other. When comparing two such hands, the hand with the higher pair is better - so for example 6-6-4-3-2 beats 5-5-A-K-Q. If the pairs are equal, compare the highest ranking odd cards from each hand; if these are equal compare the second highest odd card, and if these are equal too compare the lowest odd cards. So J-J-A-9-3 beats J-J-A-8-7 because the 9 beats the 8.

9. Nothing

Five cards which do not form any of the combinations listed above. This combination is often called High Card and sometimes No Pair. The cards must all be of different ranks, not consecutive, and contain at least two different suits. When comparing two such hands, the one with the better highest card wins. If the highest cards are equal the second cards are compared; if they are equal too the third cards are compared, and so on. So A-J-9-5-3 beats A-10-9-6-4 because the jack beats the ten.

Hand Ranking in Low Poker

There are several poker variations in which the lowest hand wins: these are sometimes known as Lowball. There are also 'high-low' variants in which the pot is split between the highest and the lowest hand. A low hand with no combination is normally described by naming its highest card - for example 8-6-5-4-2 would be described as '8-down' or '8-low'.

It first sight it might be assumed that in low poker the hands rank in the reverse order to their ranking in normal (high) poker, but this is not quite the case. There are several different ways to rank low hands, depending on how aces are treated and whether straights and flushes are counted.

Ace to Five

This seems to be the most popular system. Straights and flushes do not count, and Aces are always low. The best hand is therefore 5-4-3-2-A, even if the cards are all in one suit. Then comes 6-4-3-2-A, 6-5-3-2-A, 6-5-4-2-A, 6-5-4-3-A, 6-5-4-3-2, 7-4-3-2-A and so on. Note that when comparing hands, the highest card is compared first, just as in standard poker. So for example 6-5-4-3-2 is better than 7-4-3-2-A because the 6 is lower than the 7. The best hand containing a pair is A-A-4-3-2. This version is sometimes called 'California Lowball'.

When this form of low poker is played as part of a high-low split variant, there is sometimes a condition that a hand must be 'eight or better' to qualify to win the low part of the pot. In this case a hand must consist of five unequal cards, all 8 or lower, to qualify for low. The worst such hand is 8-7-6-5-4.

Deuce to Seven

The hands rank in almost the same order as in standard poker, with straights and flushes counting and the lowest hand wins. The difference from normal poker is that Aces are always high , so that A-2-3-4-5 is not a straight, but ranks between K-Q-J-10-8 and A-6-4-3-2. The best hand in this form is 7-5-4-3-2 in mixed suits, hence the name 'deuce to seven'. The next best is 7-6-4-3-2, then 7-6-5-3-2, 7-6-5-4-2, 8-5-4-3-2, 8-6-4-3-2, 8-6-5-3-2, 8-6-5-4-2, 8-6-5-4-3, 8-7-4-3-2, etc. The highest card is always compared first, so for example 8-6-5-4-3 is better than 8-7-4-3-2 even though the latter contains a 2, because the 6 is lower than the 7. The best hand containing a pair is 2-2-5-4-3, but this would be beaten by A-K-Q-J-9 - the worst 'high card' hand. This version is sometimes called 'Kansas City Lowball'.

Ace to Six

Many home poker players play that straights and flushes count, but that aces can be counted as low. In this version 5-4-3-2-A is a bad hand because it is a straight, so the best low hand is 6-4-3-2-A. There are a couple of issues around the treatment of aces in this variant.

• First, what about A-K-Q-J-10? Since aces are low, this should not count as a straight. It is a king-down, and is lower and therefore better than K-Q-J-10-2.
• Second, a pair of aces is the lowest and therefore the best pair, beating a pair of twos.

It is likely that some players would disagree with both the above rulings, preferring to count A-K-Q-J-10 as a straight and in some cases considering A-A to be the highest pair rather than the lowest. It would be wise to check that you agree on these details before playing ace-to-six low poker with unfamiliar opponents.

Selecting from more than five cards

Note that in games where more than five cards are available, the player is free to select whichever cards make the lowest hand. For example a player in Seven Card Stud Hi-Lo 8 or Better whose cards are 10-8-6-6-3-2-A can omit the 10 and one of the 6's to create a qualifying hand for low.

Poker Hand Ranking with Wild Cards

A wild card card that can be used to substitute for a card that the holder needs to make up a hand. In some variants one or more jokers are added to the pack to act as wild cards. In others, one or more cards of the 52-card pack may be designated as wild - for example all the twos ('deuces wild') or the jacks of hearts and spades ('one-eyed jacks wild', since these are the only two jacks shown in profile in Anglo-American decks).

The most usual rule is that a wild card can be used either

1. to represent any card not already present in the hand, or
2. to make the special combination of 'five of a kind'.

This approach is not entirely consistent, since five of a kind - five cards of equal rank - must necessarily include one duplicate card, since there are only four suits. The only practical effect of the rule against duplicates is to prevent the formation of a 'double ace flush'. So for example in the hand A-9-8-5-joker, the joker counts as a K, not a second ace, and this hand is therefore beaten by A-K-10-4-3, the 10 beating the 9.

Five of a Kind

When playing with wild cards, five of a kind becomes the highest type of hand, beating a royal flush. Between fives of a kind, the higher beats the lower, five aces being highest of all.

The Bug

Some games, especially five card draw, are often played with a bug. This is a joker added to the pack which acts as a limited wild card. It can either be used as an ace, or to complete a straight or a flush. Thus the highest hand is five aces (A-A-A-A-joker), but other fives of a kind are impossible - for example 6-6-6-6-joker would count as four sixes with an ace kicker and a straight flush would beat this hand. Also a hand like 8-8-5-5-joker counts as two pairs with the joker representing an ace, not as a full house.

Wild Cards in Low Poker

In Low Poker, a wild card can be used to represent a card of a rank not already present in the player's hand. It is then sometimes known as a 'fitter'. For example 6-5-4-2-joker would count as a pair of sixes in normal poker with the joker wild, but in ace-to-five low poker the joker could be used as an ace, and in deuce-to-seven low poker it could be used as a seven to complete a low hand.

Lowest Card Wild

Some home poker variants are played with the player's lowest card (or lowest concealed card) wild. In this case the rule applies to the lowest ranked card held at the time of the showdown, using the normal order ace (high) to two (low). Aces cannot be counted as low to make them wild.

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Double Ace Flush

Some people play with the house rule that a wild card can represent any card, including a duplicate of a card already held. It then becomes possible to have a flush containing two or more aces. Flushes with more than one ace are not allowed unless specifically agreed as a house rule.

Natural versus Wild

Some play with the house rule that a natural hand beats an equal hand in which one or more of the cards are represented by wild cards. This can be extended to specify that a hand with more wild cards beats an otherwise equal hand with fewer wild cards. This must be agreed in advance: in the absence of any agreement, wild cards are as good as the natural cards they represent.

Incomplete Hands

In some poker variants, such as No Peek, it is necessary to compare hands that have fewer than five cards. With fewer than five cards, you cannot have a straight, flush or full house. You can make a four of a kind or two pairs with only four cards, triplets with three cards, a pair with two cards and a 'high card' hand with just one card.

The process of comparing first the combination and then the kickers in descending order is the same as when comparing five-card hands. In hands with unequal numbers of cards any kicker that is present in the hand beats a missing kicker. So for example 8-8-K beats 8-8-6-2 because the king beats the 6, but 8-8-6-2 beats 8-8-6 because a 2 is better than a missing fourth card. Similarly a 10 by itself beats 9-5, which beats 9-3-2, which beats 9-3, which beats a 9 by itself.

Ranking of suits

In standard poker there is no ranking of suits for the purpose of comparing hands. If two hands are identical apart from the suits of the cards then they count as equal. In standard poker, if there are two highest equal hands in a showdown, the pot is split between them. Standard poker rules do, however, specify a hierarchy of suits: spades (highest), hearts, diamonds, clubs (lowest) (as in Contract Bridge), which is used to break ties for special purposes such as:

• drawing cards to allocate players to seats or tables;
• deciding who bets first in stud poker according to the highest or lowest upcard;
• allocating a chip that is left over when a pot cannot be shared exactly between two or more players.

I have, however, heard from several home poker players who play by house rules that use this same ranking of suits to break ties between otherwise equal hands. For some reason, players most often think of this as a way to break ties between royal flushes, which would be most relevant in a game with many wild cards, where such hands might become commonplace. However, if you want to introduce a suit ranking it is important also to agree how it will apply to other, lower types of hand. If one player A has 8-8-J-9-3 and player B has 8-8-J-9-3, who will win? Does player A win by having the highest card within the pair of eights, or does player B win because her highest single card, the jack, is in a higher suit? What about K-Q-7-6-2 against K-Q-7-6-2 ? So far as I know there is no universally accepted answer to these questions: this is non-standard poker, and your house rules are whatever you agree that they are. Three different rules that I have come across, when hands are equal apart from suit are:

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1. Compare the suit of the highest card in the hand.
2. Compare the suit of the highest paired card - for example if two people have J-J-7-7-K the highest jack wins.
3. Compare the suit of the highest unpaired card - for example if two people have K-K-7-5-4 compare the 7's.

Although the order spades, hearts, diamonds, clubs may seem natural to Bridge players and English speakers, other suit orders are common, especially in some European countries. Up to now, I have come across:

• spades (high), hearts, clubs, diamonds (low)
• spades (high), diamonds, clubs, hearts (low)
• hearts (high), spades, diamonds, clubs (low) (in Greece and in Turkey)
• hearts (high), diamonds, spades, clubs (low) (in Austria and in Sweden)
• hearts (high), diamonds, clubs, spades (low) (in Italy)
• diamonds (high), spades, hearts, clubs (low) (in Brazil)
• diamonds (high), hearts, spades, clubs (low) (in Brazil)
• clubs (high), spades, hearts, diamonds (low) (in Germany)

As with all house rules, it would be wise to make sure you have a common understanding before starting to play, especially when the group contains people with whom you have not played before.

Stripped Decks

In some places, especially in continental Europe, poker is sometimes played with a deck of less than 52 cards, the low cards being omitted. Italian Poker is an example. As the pack is reduced, a Flush becomes more difficult to make, and for this reason a Flush is sometimes ranked above a Full House in such games. In a stripped deck game, the ace is considered to be adjacent to the lowest card present in the deck, so for example when using a 36-card deck with 6's low, A-6-7-8-9 is a low straight.

Playing poker with fewer than 52 cards is not a new idea. In the first half of the 19th century, the earliest form of poker was played with just 20 cards - the ace, king, queen, jack and ten of each suit - with five cards dealt to each of four players. The only hand types recognised were, in descending order, four of a kind, full house, three of a kind, two pairs, one pair, no pair.

No Unbeatable Hand

In standard poker a Royal Flush (A-K-Q-J-10 of one suit) cannot be beaten. Even if you introduce suit ranking, the Royal Flush in the highest suit is unbeatable. In some regions, it is considered unsatisfactory to have any hand that is guaranteed to be unbeaten - there should always be a risk. There are several solutions to this.

In Italy this is achieved by the rule 'La minima batte la massima, la massima batte la media e la media batte la minima' ('the minimum beats the maximum, the maximum beats the medium and the medium beats the minimum'). A minimum straight flush is the lowest that can be made with the deck in use. Normally they play with a stripped deck so for example with 40 cards the minimum straight flush would be A-5-6-7-8 of a suit. A maximum straight flush is 10-J-Q-K-A of a suit. All other straight flushes are medium. If two players have medium straight flushes then the one with higher ranked cards wins as usual. Also as usual a maximum straight flush beats a medium one, and a medium straight flush beats a minimum one. But if a minimum straight flush comes up against a maximum straight flush, the minimum beats the maximum. In the very rare case where three players hold a straight flush, one minimum, one medium and one maximum, the pot is split between them. See for example Italian Poker.

Probability Of A Pair In Poker

In Greece, where hearts is the highest suit, A-K-Q-J-10 is called an Imperial Flush, and it is beaten only by four of a kind of the lowest rank in the deck - for example 6-6-6-6 if playing with 36 cards. Again, in very rare cases there could also be a hand in the showdown that beats the four of a kind but is lower than the Imperial Flush, in which case the pot would be split.

Hand probabilities and multiple decks

The ranking order of poker hands corresponds to their probability of occurring in straight poker, where five cards are dealt from a 52-card deck, with no wild cards and no opportunity to use extra cards to improve a hand. The rarer a hand the higher it ranks.

This is neither an essential nor an original feature of poker, and it ceases to be true when wild cards are introduced. In fact, with a large number of wild cards, it is almost inevitable that the higher hand types will be the commoner, not rarer, since wild cards will be used to help make the most valuable type of hand from the available cards.

Mark Brader has provided probability tables showing the frequency of each poker hand type when five cards are dealt from a 52-card deck, and also showing how these probabilities would change if multiple decks were used.

POKER PROBABILITIES

• Texas Hold'em Poker
  Texas Hold'em Poker probabilities
• Omaha Poker
  Omaha Poker probabilities
• 5 Card Poker
  5 Card Poker probabilities

POKER CALCULATOR

• Poker calculator
  Poker odds calculator

POKER INFORMATION

• Poker hand rankings
  Ranking of poker hands

In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

List Of Poker Winning Hands

Frequency of 5-card poker hands

The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)

The nCr function on most scientific calculators can be used to calculate hand frequencies; entering ​nCr​ with ​52​ and ​5​, for example, yields as above.

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.

Probability Of Winning Poker Hands Free

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 3♦ 7♦ 8♦ Q♥ A♥ 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 3♦ 7♣ 8♦ Q♥ A♥ 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.

Derivation of frequencies of 5-card poker hands

of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

• Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
  • Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is

    or simply . Note: this means that the total number of non-Royal straight flushes is 36.

• Four of a kind — Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. Thus, the total number of four-of-a-kinds is:

Poker Hand Probabilities Explained

• Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. Thus, the total number of full houses is:

• Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:

• Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:

• Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:

• Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:

• Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:

• No pair — A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:

• Any five card poker hand — The total number of five card hands that can be drawn from a deck of cards is found using a combination selecting five cards, in any order where n refers to the number of items that can be selected and r to the sample size; the '!' is the factorial operator:

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

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