# Poker Hand Ranges Explained

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

What does hand reading have to do with hand combinations you might ask? Well, poker is a game of deduction and to be a good hand reader, you need to be good at correctly ranging your opponents. Once you have assigned them a range, you will then need to start narrowing that range down. Combinatorics is one of the ways we do this. Putting all of this together, we obtain the following ranking of poker hands: Poker Hand Number of Ways to Get This Probability of This Hand Royal Flush 4 0.000154% Straight Flush 36 0.00139% Four of a Kind 624 0.0240% Full House 3,744 0.144% Flush 5,108 0.197% Straight 10,200 0.392% Three of a Kind 54,912 2.11% Two Pairs 123,552 4.75% One Pair 1,098,240 42.3% Nothing 1,302,540 50.1% Wait, how did I compute the probability of getting “Nothing”?

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

Poker HandDefinition
1Royal FlushA, K, Q, J, 10, all in the same suit
2Straight FlushFive consecutive cards,
all in the same suit
3Four of a KindFour cards of the same rank,
one card of another rank
4Full HouseThree of a kind with a pair
5FlushFive cards of the same suit,
not in consecutive order
6StraightFive consecutive cards,
not of the same suit
7Three of a KindThree cards of the same rank,
2 cards of two other ranks
8Two PairTwo cards of the same rank,
two cards of another rank,
one card of a third rank
9One PairThree cards of the same rank,
3 cards of three other ranks
10High CardIf 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 HandCountProbability
2Straight Flush400.0000154
3Four of a Kind6240.0002401
4Full House3,7440.0014406
5Flush5,1080.0019654
6Straight10,2000.0039246
7Three of a Kind54,9120.0211285
8Two Pair123,5520.0475390
9One Pair1,098,2400.4225690
10High Card1,302,5400.5011774
Total2,598,9601.0000000

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2017 – Dan Ma

Texas hold’em is a game of information availability – and when you go through a hold’em hand, regardless of whether it’s fixed-limit, pot-limit or no-limit hold’em, you need to be able to read the board and understand what it’s telling you. This is critically important.

The first step is to evaluate the composition of the board. This means working out what hand you hold based on your two hole cards and those on the board. But it’s also important that you connect the dots and have an awareness of what the best possible hand (the nuts) might be. Knowing what the nuts is on any given board is second nature to seasoned poker players, but it’s not always so obvious to beginners. Hopefully by the end of this lesson it will be.

## Common Flop Textures

The flop is a defining moment in hold’em and can be made up of many different card combinations. Here are a few examples of common flop textures you will see when playing hold’em:

### Poker Hand Ranges Explained Symbols

A rainbow flop means all three cards are of different suits. A rainbow flop means nobody can hold a flush without drawing on both the turn and the river, as the maximum number of suited cards a player could have at this point is three (two in the hole plus one on the board). If the turn is a card of the fourth suit, then a flush defintely won’t be possible.

### The Flush Draw Flop

This flop has two cards of the same suit (spades), which means the maximum number of suited cards a player could have at this point is four. If a player has four suited cards then he’s said to be “on a flush draw”, and could have two opportunities for hitting a flush – on the turn or on the river.

### The Suited Flop

If the flop contains three suited cards then someone could already have a made flush, with the two cards in their hand. These types of flops are very dangerous if you have hit part of it, but not the flush. Your hand is also under threat by players who might have just one spade in their hand and are drawing to a flush.

### The Paired Flop

Whenever the board shows a pair, the possible hands available increases to include full houses and four of a kinds. Therefore you should immediately realize that your opponent’s could be holding these big hands.

There are many more types of flops, such as trip flops, straight flops, and so on. The important thing to remember is that you analyze the texture of the flop at all times. It’s free information and is available for all to see – so don’t ignore it. In fact, the texture of the flop should heavily influence how you play a hand.

In order to be properly prepared for playing poker it’s essential that you can read the board and work out your best possible hand. It’s easy to do, yet even experienced players make mistakes and mis-read boards from time to time.

Example #1

The best hand you can make here would for a full house.

Example #2

In this example you’re best hand is for a King-high flush.

Example #3

In this example your best hand is for a straight. You could use one of the 4′s in your hand but it doesn’t make any difference because you’re effectively “playing the board”.

## The Nuts

The term “the nuts” means you have an unbeatable hand based on the board. When playing poker you should always think about what your opponents might have, so it’s important that you can read the board to work out what the nuts might be.

See if you can work out which starting hands would give the nuts in the following examples:

Example #1

This board doesn’t contain any pairs and only two suited cards. The best possible hand would be a straight, for anyone holding 10-7 (suited or unsuited). It would give a straight of 6789T. What would be the second best hole cards? That would be 7-5, giving a lower straight of 56789.

Example #2

The nuts on this board would be four of a kind since the board is paired. Anyone holding would have quads. What would be the second nut hand? Well since there are also a pair of 6’s on the board, anyone holding would have the second nuts.

Example #3

The board isn’t paired and there are only two suited cards, so we can discount a flush or better. A straight is also impossible since there are too many gaps to fill. The best hand here would be pocket Kings, giving a set. The second best would be pocket Jacks, for a lower set.

## Understanding the Strength of Your Hand

Sometimes the board can render those two private cards that you’ve been dealt as absolutely priceless, or absolutely worthless, or somewhere in-between. You must learn to read the board and fully understand the relative strength of your hand and what potential opportunities or dangers lay ahead.

Let’s revisit the example hands we used earlier and determine how strong your hand really is.

Example #1

You have a full house, but you could be losing to players with the following hole cards: TT, JJ, JT, T3, T2, and 33. So while you have a full house, your hand is only the eighth best hand available. You still have a strong hand, but it’s by no means the nuts.

Example #2

You have a flush but you could be losing to any player who is holding the , or . Therefore you have the fourth best hand based on this board.

Example #3

### Poker Hand Ranges Explained Signals

You are “playing the board” which shows a straight. But you would lose to any players holding a single 8 for the higher straight. Any player with an 8 would be in very big trouble if someone else had 8-9 in the hole for the nut straight.

## SWOT Analysis

In the business world it’s common to perform a SWOT analysis of a company, and SWOT stands for Strengths, Weaknesses, Opportunities, and Threats. Well, we can take this approach to poker too. Let’s look at some example starting hands and flops, and perform a SWOT analysis on them.

Example #1

• Strengths – You have flopped a set of 7’s, a very strong hand.
• Weaknesses – Well you don’t have the nuts, as this would be a player holding K-K, but you have the 2nd nuts – so there’s no real weakness at this stage.
• Opportunities – Your hand can still improve. Another 7 would give you quads, and a 5 or K on the turn or river would give a full house.
• Threats – The biggest threat to your hand is another club on the turn or river, which could make someone else a flush. While a King would give you a full house, it could also give someone else a bigger full house (i.e. if they were holding K-5 or K-7).

Example #2

• Strengths – You have flopped an open ended straight draw (5678), and a flush draw. Put simply, you’ve flopped a great drawing hand.
• Weaknesses – The weakness is that you need to hit. If the hand stopped here you have nothing but 8 high and would certainly be beaten.
• Opportunities – This hand offers great opportunities, and the best outcome would be to hit a 4 or 9 on the turn or river for the straight, as this would give you the nuts (if it isn’t a spade).
• Threats – While this hand gives a flush draw, it would be a low flush and therefore vulnerable to a higher flush. While a flush is better than a straight, the straight would place your hand in a stronger position. The lesson here is that you don’t always want the highest possible poker hand ranking for yourself, but to have one better than your opponents.

Example #3

### Poker Hand Ranges Explained Drills

• Strengths – It’s a rainbow flop and you have a pair of aces, the highest pair available.
• Weaknesses – Your kicker is weak. You could be losing to a lot of hands. Even though you hold suited cards, there is no possibility of hitting a flush.
• Opportunities – Another Ace would give you trips and another 5 would bring two pair.
• Threats – Due to the weak kicker this hand is always vulnerable and could get you into a lot of trouble. This is a prime example of why playing Ace-rag is not a good starting hand, because you never really know where you are.

## From Nuts to Nowhere!

Let’s imagine you’re playing poker and the following happens:

What’s your hand at this point? Well, you have flopped the nuts with an Ace high straight. If you were to perform a SWOT analysis at this stage it would be very healthy. The only slight danger is the potential flush draw. Let’s imagine you bet and get called, and the turn brings:

You no longer have the nuts because the board is now paired. Let’s imagine you still believe you’re winning and you get to see the river card:

This is a terrible river card. Not only does the river bring another club, meaning a possible flush, it also double pairs the board. You’re now losing if any of your opponents have just a single Queen or Jack, or two clubs. You’d also be losing if someone held TT, let alone a pair of Jacks or Queens for quads.

## Conclusion

The above hand example just goes to show how important it is to read the board and understand the strength of your own hand based on the information available to you. Remember that the more experience you have playing poker, the quicker and easier it will be to read the board and assess the relative strength of your hand.

Even experienced poker players make mistakes from time to time. As a beginner to poker you might mis-read the board every now and again – but it’s natural to make mistakes when you’re learning new skills. Just be aware of the texture of the flop at all times and use the information available to you – and use it wisely.

### Related Lessons

By Tim Ryerson

Tim is from London, England and has been playing poker since the late 1990’s. He is the ‘Editor-in-Chief’ at Pokerology.com and is responsible for all the content on the website.

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