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Texas Hold'em Poker probabilities - Omaha Poker
Omaha Poker probabilities - 5 Card Poker
5 Card Poker probabilities
There are 4 suits (Hearts, Diamonds, Clubs, and Spades). There are 13 ranks (in order: A, 2, 3., 9, 10, Jack, Queen, King). One small caveat is that Ace can stand for value 1 OR the value above the King (14). Standard 3 card poker rules: You are to make the best hand out of 3 cards. If two hands are equivalent, the tie is broken by highest.
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POKER INFORMATION
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Ranking of poker hands
/royalflush-56a73fdf3df78cf77293ac7a.jpg "Queen")
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.
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.
| Hand | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|
| Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
| Flush (excluding royal flush and straight flush) | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush) | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 1 : 1 |
Poker Straight Ace
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 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.
- 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
- 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:
- 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.
Home > 5 Card Poker probabilities
< PokerHand rankings[edit]
The most fundamental of poker concern the hand rankings, because the hand rankings determine the winner. While betting is extremely important to the game, players are wagering on whether they have won, therefore a complete understanding of hand rankings must come first. These hand rankings do not apply to games played 'low', such as lowball or razz; see the section on 'low hands' below.
The cards are ranked thus, from low to high: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace. An ace is the highest card, but it can also function as the lowest in completing a straight. The two is usually called a 'deuce', and the three is sometimes called a 'trey'. Ten, Jack, Queen, King, and Ace are often abbreviated T, J, Q, K, and A, respectively, so that each card name has a single number or letter associated with it. This is commonly used in describing hands, for example, A-2-3-4-5 is a hand with an ace, a two ('deuce'), a three, a four, and a five — not necessarily in that order, but presenting them in that order makes it clear that the hand is a straight. A hand may also be written, say, A-A-x-x-x, where 'x' means any other card that does not form a better hand.
| Rank name | Also called | Cards needed | Example | Names for example |
|---|---|---|---|---|
| High card | No pair, nothing | (Anything) | A-x-x-x-x | Ace high |
| Pair | Two cards of same rank | A-A-x-x-x | Aces; pair of aces | |
| Two pair | Two pairs | A-A-K-K-x | Aces up; aces and kings; aces over kings | |
| Three of a kind | Trips, a set | Three cards of same rank | A-A-A-x-x | Three aces; set of aces |
| Straight | Five cards in sequence | 10-J-Q-K-A | Ace-high straight | |
| Flush | All five cards same suit | A♣10♣7♣6♣4♣ | Ace-high flush | |
| Full house | Boat, full boat | Three of a kind plus a pair | A-A-A-K-K | Aces full; aces full of kings |
| Four of a kind | Quads | Four cards of same rank | A-A-A-A-x | Quad aces; four aces |
| Straight flush | Five cards forming straight and a flush | 210♠J♠Q♠K♠A♠ | Ace-high straight flush (Also called a Royal Flush) |
King Queen Poker
A-2-3-4-5 is considered a five-high straight, and it is called a wheel or bicycle; this is the only time an ace plays as a low card. An ace-high straight flush is called a royal flush and it cannot be beaten. The only time it ties is when all 5 cards to the royal flush, i.e. A♥K♥Q♥J♥10♥, are on the community board. Higher cards always beat lower cards, for example, a pair of aces beats a pair of kings, and a flush with a king beats a flush whose highest card is a Queen. If two players have the same pair, a kicker is used to break the tie if possible (more about them soon). When two players have two pair, the highest pairs are considered, for example, aces up always beats kings up, no matter the other pairs. If, for example, two players both have aces up, then the higher of the smaller pairs wins: aces over kings beats aces over queens. If, for example, both players have aces over kings, then the kicker card is considered. Kickers also come into play when more than one player has the same three or four of a kind (possible only in community card games or wildcard games). If players have the same straight, flush, full house, or straight flush, it is always a tie and the players split the pot. There is no suit superiority or trump suit; a spade flush with A-10-9-6-4 does not beat a club flush with the same values.
A kicker is any card that you hold in your hand that does not make part of it, that is, an otherwise useless card. A hand can have more than one kicker; A pair for instance has three kickers and a three-of-a-kind has two, and they are considered in rank order highest-first. When two players hold the same pair, two pair, three of a kind, or four of a kind, the highest kicker wins, for example, A-A-K-x-x beats A-A-Q-x-x, A-A-K-Q-x beats A-A-K-J-x, and A-A-K-Q-J beats A-A-K-Q-T. A kicker can be higher than the rest of the hand, for example, K-K-A-x-x beats K-K-J-x-x, so an ace usually makes the best kicker. If the first kicker ties and there is a second or third, they are compared in rank order; A-A-K-J-x loses to A-A-K-Q-x. If the hands are totally equivalent, the pot is split.
Low hands[edit]
Some games have a high-low split, and some games such as lowball or razz are played low-only. In a high-low split game, typically a low hand must not have any cards ranked higher than eight and no cards must be paired, or it does not count as a low hand. In low-only games, any cards can be used. Many forms of poker do not use low hands, so you need not concern yourself with these until you intend to play games that do.
There are three common ways of ranking low hands, ace-to-five low, ace-to-six low, and deuce-to-seven low, named after the best possible hands in the respective systems. In all systems, paired cards are bad and cannot be used to beat any hand that does not have a pair. Likewise, a pair beats three of a kind, three of a kind beats a full house, and a full house beats four of a kind. The most common hand ranking system for low hands is ace-to-five, used almost universally in high-low split games and very common in other games. This means A-2-3-4-5 (called a wheel or bicycle, just as it is as a high hand) is the best possible low hand, and the ace is the lowest card. For a high-low split game, it also forms a high hand: a five-high straight. In order to avoid confusion, we will discuss only ace-to-five low at the moment.
When pairs and any other 'bad' hands are not present, then the winner is the one whose highest card is lowest. For this reason, a low hand is usually described highest card first, to make it easier to tell which is lower. In ace-to-five, 8-4-3-2-A loses to 7-6-5-4-3 because the highest card in the first hand (eight) is higher than the highest card in the second hand (seven), even though all the other cards in the second hand are lower. If the highest cards are the same, then the next-highest cards are considered, and so on: 8-7-6-3-A loses to 8-7-5-4-2 because the second hand goes lower first.
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Texas Hold'em Poker probabilities - Omaha Poker
Omaha Poker probabilities - 5 Card Poker
5 Card Poker probabilities
There are 4 suits (Hearts, Diamonds, Clubs, and Spades). There are 13 ranks (in order: A, 2, 3., 9, 10, Jack, Queen, King). One small caveat is that Ace can stand for value 1 OR the value above the King (14). Standard 3 card poker rules: You are to make the best hand out of 3 cards. If two hands are equivalent, the tie is broken by highest.
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.
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.
| Hand | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|
| Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
| Flush (excluding royal flush and straight flush) | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush) | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 1 : 1 |
Poker Straight Ace
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 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.
- 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
- 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:
- 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.
Home > 5 Card Poker probabilities
< PokerHand rankings[edit]
The most fundamental of poker concern the hand rankings, because the hand rankings determine the winner. While betting is extremely important to the game, players are wagering on whether they have won, therefore a complete understanding of hand rankings must come first. These hand rankings do not apply to games played 'low', such as lowball or razz; see the section on 'low hands' below.
The cards are ranked thus, from low to high: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace. An ace is the highest card, but it can also function as the lowest in completing a straight. The two is usually called a 'deuce', and the three is sometimes called a 'trey'. Ten, Jack, Queen, King, and Ace are often abbreviated T, J, Q, K, and A, respectively, so that each card name has a single number or letter associated with it. This is commonly used in describing hands, for example, A-2-3-4-5 is a hand with an ace, a two ('deuce'), a three, a four, and a five — not necessarily in that order, but presenting them in that order makes it clear that the hand is a straight. A hand may also be written, say, A-A-x-x-x, where 'x' means any other card that does not form a better hand.
| Rank name | Also called | Cards needed | Example | Names for example |
|---|---|---|---|---|
| High card | No pair, nothing | (Anything) | A-x-x-x-x | Ace high |
| Pair | Two cards of same rank | A-A-x-x-x | Aces; pair of aces | |
| Two pair | Two pairs | A-A-K-K-x | Aces up; aces and kings; aces over kings | |
| Three of a kind | Trips, a set | Three cards of same rank | A-A-A-x-x | Three aces; set of aces |
| Straight | Five cards in sequence | 10-J-Q-K-A | Ace-high straight | |
| Flush | All five cards same suit | A♣10♣7♣6♣4♣ | Ace-high flush | |
| Full house | Boat, full boat | Three of a kind plus a pair | A-A-A-K-K | Aces full; aces full of kings |
| Four of a kind | Quads | Four cards of same rank | A-A-A-A-x | Quad aces; four aces |
| Straight flush | Five cards forming straight and a flush | 210♠J♠Q♠K♠A♠ | Ace-high straight flush (Also called a Royal Flush) |
King Queen Poker
A-2-3-4-5 is considered a five-high straight, and it is called a wheel or bicycle; this is the only time an ace plays as a low card. An ace-high straight flush is called a royal flush and it cannot be beaten. The only time it ties is when all 5 cards to the royal flush, i.e. A♥K♥Q♥J♥10♥, are on the community board. Higher cards always beat lower cards, for example, a pair of aces beats a pair of kings, and a flush with a king beats a flush whose highest card is a Queen. If two players have the same pair, a kicker is used to break the tie if possible (more about them soon). When two players have two pair, the highest pairs are considered, for example, aces up always beats kings up, no matter the other pairs. If, for example, two players both have aces up, then the higher of the smaller pairs wins: aces over kings beats aces over queens. If, for example, both players have aces over kings, then the kicker card is considered. Kickers also come into play when more than one player has the same three or four of a kind (possible only in community card games or wildcard games). If players have the same straight, flush, full house, or straight flush, it is always a tie and the players split the pot. There is no suit superiority or trump suit; a spade flush with A-10-9-6-4 does not beat a club flush with the same values.
A kicker is any card that you hold in your hand that does not make part of it, that is, an otherwise useless card. A hand can have more than one kicker; A pair for instance has three kickers and a three-of-a-kind has two, and they are considered in rank order highest-first. When two players hold the same pair, two pair, three of a kind, or four of a kind, the highest kicker wins, for example, A-A-K-x-x beats A-A-Q-x-x, A-A-K-Q-x beats A-A-K-J-x, and A-A-K-Q-J beats A-A-K-Q-T. A kicker can be higher than the rest of the hand, for example, K-K-A-x-x beats K-K-J-x-x, so an ace usually makes the best kicker. If the first kicker ties and there is a second or third, they are compared in rank order; A-A-K-J-x loses to A-A-K-Q-x. If the hands are totally equivalent, the pot is split.
Low hands[edit]
Some games have a high-low split, and some games such as lowball or razz are played low-only. In a high-low split game, typically a low hand must not have any cards ranked higher than eight and no cards must be paired, or it does not count as a low hand. In low-only games, any cards can be used. Many forms of poker do not use low hands, so you need not concern yourself with these until you intend to play games that do.
There are three common ways of ranking low hands, ace-to-five low, ace-to-six low, and deuce-to-seven low, named after the best possible hands in the respective systems. In all systems, paired cards are bad and cannot be used to beat any hand that does not have a pair. Likewise, a pair beats three of a kind, three of a kind beats a full house, and a full house beats four of a kind. The most common hand ranking system for low hands is ace-to-five, used almost universally in high-low split games and very common in other games. This means A-2-3-4-5 (called a wheel or bicycle, just as it is as a high hand) is the best possible low hand, and the ace is the lowest card. For a high-low split game, it also forms a high hand: a five-high straight. In order to avoid confusion, we will discuss only ace-to-five low at the moment.
When pairs and any other 'bad' hands are not present, then the winner is the one whose highest card is lowest. For this reason, a low hand is usually described highest card first, to make it easier to tell which is lower. In ace-to-five, 8-4-3-2-A loses to 7-6-5-4-3 because the highest card in the first hand (eight) is higher than the highest card in the second hand (seven), even though all the other cards in the second hand are lower. If the highest cards are the same, then the next-highest cards are considered, and so on: 8-7-6-3-A loses to 8-7-5-4-2 because the second hand goes lower first.
In ace-to-six low, straights and flushes count for high (that is, they're bad), and the best possible hand is A-2-3-4-6 unsuited, since it's the lowest possible card combination that avoids pairing, straights, and flushes. Deuce-to-seven is identical except the ace is the highest card, so the best possible hand is 2-3-4-5-7 unsuited. Therefore, in deuce-to-seven low, the hand that would make the worst possible high hand in traditional poker is the best possible low hand, and vice versa: a royal flush is the worst possible hand.
Poker Straight Queen King Ace Two Three Wheelers
