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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

However, professional poker players swear that developing certain skills, for example poker hand ranking, can help you much more than any talisman.

I agree with poker pros about the skill factor in real money poker, which is why you are reading this guide right now.

For all of you who are still making baby steps in poker, this in-depth guide is there to help you learn the necessary things to accelerate your journey. You will get the information on:

Once you go through the text, you will find the list of frequently asked questions with the straight-to-the-point answers. Therefore, if you are ready to learn all about poker hand ranking and get launched to the stars of the poker world, let’s begin.

What Makes Poker a Game of Skill?

It is a common belief that poker is a game of luck and not a game of skill. At the end of the day, poker is available in every casino, be it a land-based or an online one, among all other games for which winning is purely based on luck.

Yet, the issue was reconsidered many times, until a federal judge from Brooklyn, Jack B. Weinstein, described the one who wins at poker as “the player who could guess his opponents’ intentions and disguise his own, make calculated decisions on when to hold and fold, and quickly decide how much to wager.”

Undoubtedly, poker differs from other games like slots, roulette, bingo and various lotteries in this aspect, as well as many others that I mentioned below.

You Play Against Humans, Not Machines

Whenever experts analyze an outcome of a specific situation, the final thoughts always lead to the human factor. Regardless of the area of expertise, if poker players are in charge of a particular action, they carry the greatest responsibility.

You might wonder how this can be applied to poker, but you are about to find out that the explanation is, indeed, pretty simple. When you play slots, your opponent is a machine. Machines are easy to program and do not have the ability to think on their own.

They use programs like the Random Number Generator (RNG) to determine who is going home with pockets full of money. That’s why slots are completely based on luck and, unless the day you visit a casino is your lucky day, there’s no chance you will win any cash by playing slots.

On the other hand, when you play poker, your opponents are other humans. There’s no machine that is programmed to sabotage your game and pre-determine the winner.

Also, poker is one of the rare games where the house doesn’t have much influence. The house does collect the money from cash game pots and buy-ins, but it does not compete against players.

Therefore, you are facing flesh-and-blood competitors like yourself who are prone to making mistakes. After all, the trial-and-error system has been used for years in order to come up with some of the greatest inventions in history.

In other words, your opponents are humans who use their own head or luck to win at this game, which you can do as well. This is why you should go through this guide and learn some poker tips and tricks that will give you a significant advantage over most of the novice players you will come across.

Good Math Skills Make You Good at Poker

Math and casino games have more in common than you think. What is more, casinos often hire brilliant math graduates to carefully examine the establishment’s operation and help the house get more than a fair share of profit.

Likewise, if you are good at math, there’s no reason why you wouldn’t be able to use these skills to win some money in poker.

There’s no need to be a genius, as even simple math rules will help if used correctly. You make decisions based on the hand you’ve got and the cards you see on the table. Since you cannot see other players’s cards, you need to calculate the odds of your hand and play accordingly.

Let’s assume that you are familiar with the pack of cards and the poker hand ranking system. You hold two cards and have five more on the table. Once you decide what is the best combination out of all the cards you can use, you can move on to calculating the odds.

Obviously, no one else can have the cards you’ve got, and this is where you start from when shaping possible outcomes of the round. Sometimes it is easy to determine if your hand is the strongest one, while some hands might be tricky.

When you have a strong hand, rely on math and don’t hesitate to invest more. If your hand is not that great, do not put a lot of money on the table. Don’t forget that you can always fold if you evaluate that your chances to win are close to zero.

Also, surprises might occur at any point while you are focused on the great cards you’ve been dealt. It is possible that, sometimes, you have a strong hand mathematically, but you still lose in the end. Don’t get disappointed and keep using your strategy because it will pay off in the long run.

Bluffing and Body Language

One more thing you should always have in mind when you play poker is that players have the ability to control their body language and facial expressions. In real life, this is called acting, but in poker, we call it a bluff.

Even regular poker players who are already familiar with bluffing can be deceived from time to time. Bluff can help you win even if your chances are statistically pretty low based on your hand. However, in order to successfully bluff, you need to practice a lot and be absolutely aware of your moves and expressions.

There are many things that can reveal a bluff. One of them is a body language that needs to be perfectly controlled in order for a bluff to work. When a player decides to bluff for the first time, there will be some signs that can reveal their intentions to others. The most usual ones are heavy sweating and frequent movements with little eye contact.

The time a player takes to make a decision can also tell you that they are bluffing. When a player invests a lot after thinking too much but still doesn’t go all-in, it can mean that they are just trying to scare you off.

The time of placing a bet is also connected to the size of a bet. If a player doesn’t want to fold or go all in, then the quality of cards that the player is holding is probably average because you don’t need to think a lot if they hold extremely good or bad cards.

All in all, apart from concentrating on the cards you hold and the ones on the table, you need to take into consideration all the other aspects of the game, including other players and their behaviour.

Poker Hands Ranked from Highest to Lowest

In a game of poker, impossible can happen. You can have a strong pair but ultimately lose from someone who initially got a badly-dealt hand. Before you start developing a strategy for playing poker, you need to memorize poker hand rankings by heart.

Royal Flush

Royal Flush represents the best possible outcome in poker, and is, consequently, the hardest one to hit. If you get the combination of a 10 ♠, J ♠, Q ♠, K ♠ and A ♠ of the same suit, consider yourself extremely lucky, as this combination is unbeatable.

It is highly unlikely for two players to get the Royal Flush, but if they do, no specific suit has an advantage, and the winners split the pot.

Straight Flush

The runner-up hand is the Straight Flush. It consists of any straight of the same suit lower than the Royal Flush.

This is one of those hands that might seem useless at the beginning when, for example, you get a 5 ♦ and 6 ♦ of the same suit. However, you can end up winning that round if the cards on the table are a 7 ♦, 8 ♦ and 9 ♦ of the suit you already have in your hands.

Four of a Kind

The next hand in line is called “Four of a Kind”, where the name is self-explanatory. You have four cards of the same value for example A ♠, A ♦, A ♣ and A ♥, plus one more card known as the kicker for example, 9 ♣ .

Logically, when two or more players have this hand, the one with the highest value wins. The kicker is used in situations when two players have Four of a Kind of the same rank. In that case, the player with the higher kicker takes all the money.

Full House

Full House is a hand where you’ve got a combination of three cards of the same rank for example Q ♠, Q ♦, Q ♥ and one pair of a different rank for example J ♣ and J ♦.

It can be comprised of cards of any rank, but in the event of a tie, the highest three matching cards are the winning hand.

Flush

To get a Flush, it is necessary to have five cards of the same suit. They don’t need to be in consecutive order or to include the highest cards for example A ♥, J ♥, 9 ♥, 6 ♥ and 3 ♥.

Yet, it is possible that someone else might have a flush as well, in which case the suit won’t matter at all. What will matter is your highest card and, if it’s still a tie, the second-highest, third-highest, fourth-highest, and fifth-highest card.

Straight

Straight stands right behind the Flush and features any five cards in consecutive order, regardless of the suit for example 5 ♠, 6 ♦, 7 ♠, 8 ♦ and 9 ♣.

A player can decide whether an Ace will be perceived as a high or a low card, depending on the combination of other cards. The highest possible straight is victorious in the case of a tie.

Three of a Kind

When you’ve got three cards of the same rank, you have three of a kind for example 2 ♠, 2 ♦ and 2 ♥. Although this is not the most desirable hand one could get, it can bring you some money in rounds where no one seems to be lucky enough and get better cards.

The best Three-of-a-Kind is the one with Aces, and it becomes less powerful as the rank goes down.

Two Pair

Two Pair hand does not require further explanation, but to be on the safe side, I will still give you one. This is a hand in which you have a pair of cards belonging to the same rank, as well as another pair of cards of another rank for example A ♥, A ♠, K ♥ and K ♠.

Image Of Poker Hands

When someone else playing at your table also has this hand, whoever has the strongest first pair will win. If, by any chance, you’ve got the same pair, then the other pair jumps in to assist you. If you have a higher second pair, you win. However, if you are one the same page, then the fifth card determines the winner.

One Pair

One Pair is, simply put, a pair of cards of the same rank for example A ♦ and A ♠. The highest pair wins in a situation where another person also has one pair.

If the rank of the pairs is the same, the highest side card brings money home. When the first kicker is the same as well, the second one, and even the third one might become the one.

High Card

Whatever you’ve got and cannot find on the list provided above falls under the category known as a High Card for example, you get a 5 ♠, 6 ♦, J ♣, Q ♦ and A ♠.

Many would say that this hand basically means you have nothing to bet on, but if other players are as unlucky as you are, good high cards can be beneficial.

Poker Hand Ranking Calculator

This guide is probably unnecessary to all the poker pros who are in need of some more advanced tips and tricks. Yet for all of you who still struggle to decide whether you have a chance to win a round, there’s a surprise.

The first thing I would like to mention is that practice makes perfect. The more you play, the better you will get as time goes by. However, your bank account would agree that it would be much better to practice without investing real money, at least it the beginning.

Nowadays, many online casinos offer demo versions of the majority of their games, including poker. This allows you to play a game for some time until you are confident enough to play against real players and put some money in the picture.

Another great option that you can use to master poker hand rankings is a poker hand ranking calculator. Plenty of them can be found online, and they are very simple to use. You can type in all the information you have regarding your cards and, if you are playing amateur poker at home, the cards of your opponents. The calculator will check all the combinations and let you know if your decision was the right one.

Conclusion

Playing poker just for fun is great, but if you are interested in some real money, you need to put in some effort. Learning poker hand ranking is not as hard as it might seem at the beginning, and with little practice, it can take you a long way.

Once you go through all the poker hands, you will see that poker really is a game of skill. I hope that you found this article useful and easy to comprehend. In case you have any further questions, you can proceed to the FAQ section, or write it down in the comment section below. If not, I wish you lots of luck in your future poker games.

FAQs

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