- Simply put, a poker hand is just a collection of specific cards. At the end of each game (i.e., the showdown stage), all active players must compare their respective hands to determine a winner (or winners in case there’s an unbreakable tie).
- Knowing what beats what in poker or Texas Hold’em is an important early step in learning the game. To help you out, I have provided for you an attractive printable or downloadable “cheat sheet” for both 5 card hand rankings as well as top 24 pre-flop starting hands.
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.
The strongest poker hand is the royal flush. It consists of Ten, Jack, Queen, King, and Ace, all of the same suit, e.g. Diamonds, spades, hearts, or clubs. #2 Straight Flush The second strongest hand in poker is the straight flush. Poker Hand Combinations Explained. Poker hands fall into one of ten categories. The highest is a royal flush, followed by a straight flush, then four of a kind, a full house, a flush, a straight. This hand has any five cards, all of the same suit, in order, such as the 4-5-6-7-8 all of hearts. It doesn't bring quite the adrenaline rush of a royal flush, but it's still a reliable winner.
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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.
Order Of Best Hands In Poker
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
| Poker Hand | Definition | |
|---|---|---|
| 1 | Royal Flush | A, K, Q, J, 10, all in the same suit |
| 2 | Straight Flush | Five consecutive cards, |
| all in the same suit | ||
| 3 | Four of a Kind | Four cards of the same rank, |
| one card of another rank | ||
| 4 | Full House | Three of a kind with a pair |
| 5 | Flush | Five cards of the same suit, |
| not in consecutive order | ||
| 6 | Straight | Five consecutive cards, |
| not of the same suit | ||
| 7 | Three of a Kind | Three cards of the same rank, |
| 2 cards of two other ranks | ||
| 8 | Two Pair | Two cards of the same rank, |
| two cards of another rank, | ||
| one card of a third rank | ||
| 9 | One Pair | Three cards of the same rank, |
| 3 cards of three other ranks | ||
| 10 | High Card | If no one has any of the above hands, |
| the player with the highest card wins |
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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.
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
| Poker Hand | Count | Probability | |
|---|---|---|---|
| 2 | Straight Flush | 40 | 0.0000154 |
| 3 | Four of a Kind | 624 | 0.0002401 |
| 4 | Full House | 3,744 | 0.0014406 |
| 5 | Flush | 5,108 | 0.0019654 |
| 6 | Straight | 10,200 | 0.0039246 |
| 7 | Three of a Kind | 54,912 | 0.0211285 |
| 8 | Two Pair | 123,552 | 0.0475390 |
| 9 | One Pair | 1,098,240 | 0.4225690 |
| 10 | High Card | 1,302,540 | 0.5011774 |
| Total | 2,598,960 | 1.0000000 |
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2017 – Dan Ma
Every poker player knows that the Royal Flush is the strongest poker hand, but where do all of the other poker winning hands rank? Here is a comprehensive list of poker hands in order from highest to lowest ranking. If you are new to the game of poker, learning the different poker hands is a great first step in learning how to beat your opponents with the cards you are dealt.
#1 Royal Flush
The strongest poker hand is the royal flush. It consists of Ten, Jack, Queen, King, and Ace, all of the same suit, e.g. diamonds, spades, hearts, or clubs.
#2 Straight Flush
The second strongest hand in poker is the straight flush. It is composed of five consecutive cards of the same suit. If two players have a straight flush, the player with the highest cards wins.
#3 Four-of-a-kind
Different Poker Hands In Order Of Best To Worst
A four-of-a-kind is four cards of the same rank, e.g. four Aces. If two players have four-of-a-kind, then the one with the highest four-of-a-kind wins. If they have the same (if four-of-a-kind is on the board), then the player with the highest fifth card wins, since a poker hand is always composed of five cards.
#4 Full House
A full house is a combination of a three-of-a-kind and a pair. If two players have a full house, then the one with the highest three-of-a-kind wins. If they have the same one, then the pair counts.
#5 Flush
Five cards of the same suit make a flush. If two players have a flush, then the one with the highest cards wins.
#6 Straight
Five consecutive cards are called a straight. If two players have a straight, the one with the highest cards wins.
#7 Three-of-a-kind
A three-of-a-kind is composed of three cards of the same rank. If two players have the same three-of-a-kind, then the other cards, or both cards, determine the winner, since a poker hand is a always composed of five cards.
#8 Two-pair
Two-pair hands are, of course, composed of two pairs. If two players have two-pair, the rank of the higher pair determines the winner. If they have the same higher pair, then the lower one counts. If that is also the same, then the fifth card counts.
#9 Pair
A pair is composed of two cards of the same rank. Since a poker hand is always composed of five cards, the other three cards are so-called “kickers”. In case two players have the same pair, then the one with the highest kicker wins.
#10 High card
If you don’t even have a pair, then you look at the strength of your cards. If there are two players at showdown who don’t have a pair or better, then the one with the highest cards wins.
Any of the PalaPoker.com games use the standard rank of hands to determine the high hand.
However, at PalaPoker.com we also play “split pot” games, like Omaha Hi-Lo8 and Stud Hi-Lo8, in which the highest hand splits the pot with a qualifying (“8 or better”) low hand; therefore, we must also be familiar with:
Low Poker Hands List:
This method of ranking low hands is used in traditional Hi/Lo games, like Omaha Hi/Lo and Stud Hi/Lo, as well as in Razz, the ‘low only’ Stud game.
Note that suits are irrelevant for Ace to Five low. A flush or straight does not ‘break’ an Ace to Five low poker hand. Aces are always a ‘low’ card when considering a low hand.
Please also note that the value of a five-card low hand starts with the top card, and goes down from there.
#1 Five Low, or “Wheel“: The Five, Four, Three, Deuce and Ace.
In the event of a tie: All Five-high hands split the pot.
#2 Six Low: Any five unpaired cards with the highest card being a Six.
In the event of a tie: The lower second-highest ranking card wins the pot. Thus 6,4,3,2,A defeats 6,5,4,2,A. If necessary, the third-highest, fourth-highest and fifth-highest cards in the hand can be used to break the tie.
#3 Seven Low: Any five unpaired cards with the highest card being a Seven.
In the event of a tie: The lower second-highest ranking card wins the pot. If necessary, the third- highest, fourth-highest and fifth-highest cards in the hand can be used to break the tie.
#4 Eight Low: Any five unpaired cards with the highest card being an Eight.
In the event of a tie: The lower second-highest ranking card wins the pot. If necessary, the third-highest, fourth-highest and fifth-highest cards in the hand can be used to break the tie. An Eight Low is the weakest hand that qualifies for low in Omaha Hi/Lo and Stud Hi/Lo.
Check back here as you are learning the game of poker for a list that details the poker hands order. Sign up today to start winning real money!