Poker Ranking Of Hands Chart

As shown in the poker hand rankings chart, the order of poker rankings (from the highest to the lowest) is: Royal Flush, Straight Flush, Four-of-a-Kind, Full House, Flush, Straight, Three-of-a. 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.

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.

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

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2017 – Dan Ma

Almost all variants of poker are based around a poker hand that’s made up of 5 cards. It is these 5 cards that determines who the winner is. Therefore, it is critical that you understand the ranking order of poker hands so that you know how strong your poker hand is compared to your opponents.

Royal Flush

This is the best possible hand you can get in poker and is unbeatable. It’s basically a “Straight Flush” that runs from Ace to Ten.

Straight Flush

Five cards in numerical order and are all of the same suit. This is similar to a “Royal Flush” with the only exception being that it does not contain an Ace. In instances where multiple players have a Straight Flush, the one with the highest high value card wins.

Four of A Kind (also known as “Quads”)

As the name suggests, it’s a combination of four cards of the same rank and a kicker (the fifth card).
In the event there are more than 1 player with this hand, the one comprised of the highest quads wins.

In community card games, where players can use cards from the “board” (cards are placed faced up, and can be used by any player to create a best 5-card hand), and therefore it’s possible that multiple player can have “Four of a Find” made up from the same quads, then the player with the highest kicker wins.

Full House

Fourth in the poker hand rankings is the Full House. It is a combination of any three cards of the same numerical value and a pair of another value. When there’s multiple “Full House” hands, the one with the highest triplet wins. If players have the same triplets, then the one with the highest pair wins.

Flush

A “Flush” is a set of five cards with the same suit. The cards do not have to be in sequential order be a Flush. Between two or more flushes, the one with the highest high value card wins, with an “Ace-high Flush” being the strongest.

In the event that multiple hands have the same highest high value card, the winner is then determined by the second highest high value card, so on and so forth.

Straight

Any five cards in sequential order that is of different suits. Two Straights are compared by the value of their high cards. An ace can be used as the highest value card in the Straight, or the lowest value card in a Straight. For example, an Ace-high straight, like the one in our example, is the strongest Straight you can get (also known as a “Broadway” Straight). But a Straight consisting of Ace, two, three, four, five is considered the weakest Straight (also known as “Wheel” straight).

Three of a Kind (also known as “Trips” or “Set”)

A three of a kind is just a Full House without the additional pair. Meaning, the other 2 cards are of different values. When there are multiple “Three of a Kind”, the one with the highest triplet wins.

If players have the same triplets, then the winner is determined by the one with the highest value of the two remaining cards, and if multiple players have the same card here again, then the player with the second highest high value card wins.

Cached

Two Pairs

You probably can guess what this is. It’s simply Two Pairs of any value. In instances where there are more than 1 player with a Two Pairs, the winner is first determined by the one with the highest pair, then the highest second pair, then the kicker (fifth card).

Poker Ranking Of Hands Charts

One Pair

This hand consists of one pair of the same value cards, and 3 unrelated cards, all of different value. In the event of a tie, the hand with the highest pair wins. If players have the same highest pair, then the winner is determined by the hand with the highest value card of the 3 kickers, then the 2nd highest, then then 3rd.

High Card

Poker Hand Rankings - Card Player

If your hand doesn’t fall into any of the above categories, then what you have is a “High Card” hand and the value of the highest card in the hand determines the strength of this hand. So, a player with a 10 High Card would beat a player with a 8 High Card.

Poker Ranking Of Hands Chart Images

In instances where multiple players have the same highest high card, then it goes down the remaining four cards to determine who the winner is.

As shown in our example, the strongest High Card hand is a A, K, Q, J, 9.

Frequently Asked Questions

Yes. The ranking order is absolute, and there are no instances where a lower ranked hand would beat a higher ranked hand.

Your friend is wrong. Suits don’t play a role in determine the strength of a hand in poker. Only the numerical value of the card does. So in this example, since you both have the same two pairs, then the winner would be the one with the highest kicker (the fifth card). So, that would be you, because your 10 is higher than your friend’s 9.

No, it doesn’t. As we explained in the above question, suits play no roles in poker. This is same for the colour of a card.

Even Though there are seven cards in total, you are only able to use 5 cards from the seven to create your strongest 5-card hand. Therefore, it’s not possible to have 3 pairs, which requires 6 cards. In this particular instance, what you have is a Two Pair hand.

No, it doesn’t. Aces can only be used as a high card or a low card in the case of any poker hand that requires the numbers to be in a sequential order, such as a “Straight” or a “Straight Flush”. So, a “10, J, Q, K, A” is the highest Straight possible (also known as a “Broadway” Straight), and a “A, 2, 3, 4, 5” is the lowest Straight possible (also known as a “Wheel” Straight). But Aces cannot be used as a wraparound Straight, such as “K, A, 2, 3, 4”.