High card by suit and low card by suit refer to assigning relative values to playing cards of equal rank based on their suit. When suit ranking is applied, the most common conventions are:

These are standard hand rankings for most poker games and apply to all high-hand poker variations including Texas Hold'em, Omaha and Stud. You'll find a printable poker hand rankings chart below the hand rankings as well as answers to some of the most frequently asked questions about poker hands and poker hand ranking. A straight flush is five consecutive cards of the same suit. If two players have a straight flush then the highest card wins. The highest possible straight flush, and the best hand in poker, is an ace high straight flush, also known as a. The absolutely best hand in poker is the Royal Flush, which is made of Ace, King, Queen, Jack, and Ten, and all of those must be the same suit. For Example, Ah Kh Qh Jh Th would make a Royal Flush hand and is unbeatable by any other combination. A 2-7 offsuit hand is the worst hand to start with in Texas Hold 'Em poker because there are so few good options available to you: you have no straight draw, no flush draw, and even if you wind up with a pair of 7s or a pair of 2s, you're unlikely to have the best hand. Of course, you'll see some crazy flops every now and then, but just because. An ace-high straight flush, commonly known as a royal flush, is the best possible hand in many variants of poker. In poker, players form sets of five playing cards, called hands, according to the rules of the game. Each hand has a rank, which is compared against the ranks of other hands participating in the showdown to decide who wins the pot.

• Alphabetical order: clubs (lowest), followed by diamonds, hearts, and spades (highest). This ranking is used in the game of bridge.
• Alternating colors: diamonds (lowest), followed by clubs, hearts, and spades (highest). Similar to alphabetical ranking in that the two highest rankings are occupied by the same two suits (hearts and spades) in the same relative position to one another, but differing in the two lowest rankings, which while occupied by the same two suits (clubs and diamonds) have their relative position to one another swapped. This ranking is sometimes used in the Chinese card game Big Two or Choh Dai Di.
• Some Russian card games like Preference, 1000 etc. use the following order: spades (lowest), clubs, diamonds and hearts (highest). The Australian card game 500 also uses this ordering.
• Some German card games (for example Skat) use the following order: diamonds (lowest), hearts, spades and clubs (highest).

Poker[edit]

Most poker games do not rank suits; the ace of clubs is just as good as the ace of spades. However, small issues (such as deciding who deals first) are sometimes resolved by dealing one card to each player. If two players draw cards of the same rank, one way to break the tie is to use an arbitrary hierarchy of suits. The order of suit rank differs by location; for example, the ranking most commonly used in the United States is not the one typically used in Italy.

Cards are always compared by rank first, and only then by suit. For example, using the 'reverse alphabetical order' ranking, the ace of clubs ranks higher than any king, but lower than the ace of diamonds. High card by suit is used to break ties between poker hands as a regional variance,^[1] but more commonly is used in the following situations, as well as various others, based upon the circumstances of the particular game:

• Randomly selecting a player or players.

To randomly select a player to deal, to choose the game, to move to another table, or for other reasons, deal each player one card and the player with high card by suit is selected. Multiple players can be selected this way.

• Assigning the bring-in.

In games such as Seven-card stud, where the player with the lowest-ranking face-up card is required to open the first betting round for a minimal amount, ties can be broken by suit. In such low stud games as razz, the player with the highest-ranking upcard must post the fractional bet.

• Awarding odd chips in a split pot.

In High-low split games, or when two players' hands tie, the pot must be split evenly between them. When there is an odd amount of money in the pot that can't be split evenly, the odd low-denomination chip can be given to the player whose hand contains the high card by suit. (This solution is not necessary in games with blinds, in which case the odd chip between high and low is awarded to the high hand, and the odd chip between a split high or split low is awarded to the first player following the dealer button.)

• Breaking ties in a chip race

During poker tournaments, a chip race is used to 'color up' large numbers of smaller-denomination chips, and a modified deal is used to assign leftover chips. Ties in the deal are broken by suit.

Contract bridge[edit]

In bridge, suit rank during the bidding phase of the game is by ascending alphabetical order.

During the play of the cards, the trump suit is superior to all other suits and the other suits are of equal rank to each other. If there is no trump suit, all suits are of equal rank.

References[edit]

1. ^'Rules of Card Games: Poker Hand Ranking'. www.pagat.com. Archived from the original on 28 May 2010. Retrieved 24 April 2018.

External links[edit]

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.

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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.

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.

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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 Hands

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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.

What Is The Best Hand In Poker

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 Hands

Highest Poker Hand Suit

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2017 – Dan Ma