Delving Into Poker Combinatorics!

Combinatorics in the context of poker deals with calculating the exact number of combinations different player hands present in specific game situations.

• Paired hands consist of 6 unique combinations for each hand:
  
  
  
• Unpaired hands can comprise 16 different combinations:
  
  • Out of those combinations, 4 will be suited, with one representing each of the four suits: diamonds, clubs, spades, and hearts.
    
    
    
  • The remaining 12 combinations will be unsuited, meaning they include cards from different suits.
    
    

In this write-up, we aim to assist you in mastering combinatorics in poker, focusing on these key topics:

Why Is Combinatorics Vital in Poker?

Many people say that poker is a maths-based game This assertion holds a significant degree of truth!

Thus, grasping and applying combinatorics in poker can dramatically enhance your mathematical approach to the game:

What Is the Total Number of Hand Combinations?

The following grid illustrates that there are 169 distinct poker hands:

1. Paired Hands are signified by the white diagonal line stretching from the top-left corner to the bottom-right.
2. Suited Unpaired Hands are found in the upper-right section of the grid, indicated by cyan hands.
3. Unsuited Unpaired Hands are depicted in the lower-left area of the grid, shown in blue.

Here is another grid that reiterates the hands while also detailing the number of combinations for each hand.

From the charts presented, we can extract the following data: poker combinatorics :

Combinatorics and Hand Range Charts

With this knowledge, our comprehension of the game can improve significantly. For instance, when building ranges, we can now incorporate combinatorics into our calculations.

• This enables us to create accurate visual representations aimed at capturing X% of hands within a specific range.

For instance, let's examine an example of an opening range from the Button position in a 100bb 6-max cash game:

By counting the number of different hands within that range, we find there are 534 unique combinations highlighted. (This also corresponds with a feature in our poker software that reveals this data.)

Given that there are 1,326 potential holdings in total, the range outlined above corresponds to approximately 40% of our starting hands. hand range !

Let's consider another scenario. What would 80% of hands represent? (This percentage often reflects a standard Small Blind opening strategy in heads-up play.)

How does this information influence your decisions moving forward?

Implementing Combinatorics at the Lower End of Hand Ranges

What percentage of hands should we typically present from each position when we are the first to act? The common answers are generally known—15% from UTG, 40% from the BTN, and so forth.

However, the specific hands that comprise this percentage largely depend on your strategic preferences!

Naturally, you will always include the strongest hands in your openings. Yet, as you approach the lower end of your opening range, you have room for creativity :

• Would you prefer more combinations of pocket pairs?
• Do you want to see a higher proportion of suited hands compared to unsuited ones?
• Would you like to half the count of certain hands to allow for a greater variety of potential hands? (For example, limiting yourself to only 6 unsuited combos instead of all 12 could permit the inclusion of an extra hand.) pocket pair and still satisfy the right %.)

This variability explains why starting hand charts often differ across various educational materials. They usually showcase a similar percentage of hands from each position.

Yet, there is flexibility in how the hands are distributed toward the lower tiers of the range.

An average button opening range typically encompasses around 40% of hands. Suppose the players in the small blind are quite loose. big blind are incredibly tight.

In that case, you might find yourself inclined to open from 50% of hands. It’s even possible to extend that range to include 100% of hands, depending on the tightness of your opponents.

What Role Does Combinatorics Play in Bluffing Frequencies?

What Role Does Combinatorics Play in Bluffing Frequencies?

Let’s suppose, after cbetting After progressing through the flop and turn, you now find yourself on the river with a failed straight draw. (This scenario can often serve as an ideal opportunity for bluffing in a triple-barrel situation.)

Before deciding to bluff with this specific busted straight draw, you should consider several questions:

1. How many hands are you planning to value-bet ?
2. What total number of failed straight draw combinations do you possess?
3. What is the distribution of suited versus unsuited combinations for that particular hand?

Let’s take a closer look at some useful tips concerning each question:

1. As a general guideline, aim for about 2 to 2.5 value hands for every 1 bluff you intend to make on the river. Therefore, if you have identified 40 value combos for value betting, ideally, you should limit yourself to around 15 to 20 bluff combos.
   
2. Consider a board displaying Q-J-7-6-2. A multitude of straight draws may have continued to bet on both the flop and turn:
   
   AK, AT, KT, T9, T8, 98, 95, 85, 84, 54, 53, 43
   
   Assuming access to all combinations, you would have a total of 12 x 16 = 192 potential bluffing combos! You’ll either need to identify over 400 value betting hands (which is generally unattainable!), or you'll have to be very selective about your bluffing choices.
   
   The best candidates are usually those that block the potential value hands your opponent might hold. (For instance, KT would prevent your opponent from having KQ and QT, both of which are hands likely to call you, and it’s preferable for them not to have those.)

Understanding Poker Hand Probabilities through Combinatorics

In this part of the text, we will explore some quick and straightforward poker combinatorics. These figures illustrate the numerous ways to construct some of the more powerful hands in the game of poker.

Poker Probability: Combinatorics of Four of a Kind

When it comes to paired boards, there exists only a single combination that can form quads. For instance, consider the scenario where the A♣ and A♠ land on the flop.

• Only one hole-card combo (A♦A♥) remains for players to make quads.

However, if there’s already three-of-a-kind present on the board, this increases the number of possible quad combinations significantly.

To calculate the number of combinations, take the one necessary 'quad card' and multiply it by the number of unknown cards left in the deck (which would be 47 after the flop). This demonstrates that there are many more combinations available beyond just the one with paired hole cards.

(It's important to note that you likely won't possess all of these combinations in your opening hand range. Yet, this provides insight into the heightened probability of achieving quads when there are three-of-a-kind on the board.)

Poker Probability: Combinatorics of Full Houses

When contemplating boats, picture a paired board such as A-Q-8-8-3. Here are the combinations of hands that can create full houses:

• A8 (6 combos) ==> (2 remaining 8’s in the deck) x (3 remaining A’s)
• Q8 (6 combos)
• 83 (6 combos)
• AA (3 combos)
• QQ (3 combos)
• 33 (3 combos)

Thus, there are 27 possible combinations of full houses on this board. Nevertheless, not all of these possible combinations may fall within a player's hand range. This is influenced by their position at the table and the actions taken during the hand.

• For instance, a player who called preflop is unlikely to have AA or QQ in their hand range, which eliminates 6 potential boat combinations immediately. They may also not call on both the flop and turn with pocket 3’s if there was any betting on those streets.

Moreover, 83 (whether suited or not) typically won’t be included in most players' hand ranges. Similarly, combinations like Q8 and perhaps A8 may not consist of all possible off-suit combos in a player's initial hand range.

It's crucial to analyze which player has more of these 'strong hand' combinations accessible within their range. This analysis helps in assessing who holds a narrower range and/or a nut advantage .

Poker Probability: Combinatorics of Flushes

When calculating combinatorics related to flushes, the situation becomes intriguing. It can be quite difficult to accurately determine how many suited hand combinations exist in a player’s range. (Some players in live games choose to play ALL of their suited hands!). Furthermore, the way these hands are split across different betting lines post-flop proves complicated to analyze.

Regardless, let’s say that the community cards reveal 3 cards of the same suit. This leaves 10 additional cards in that suit available to assist in forming a flush. flush .

To illustrate, if we take 10 cards and multiply by 9 cards, then divide by 2 (due to drawing), we find there are 45 various potential combinations available once the 3-flush lands on the board (using A-K-Q all in the same suit, for this example):♥7♥is the same as 7♥8♥As indicated earlier, not all 45 flush combinations may be present in the opponent’s range. For instance, consider a hypothetical 6max UTG open range that comprises only 15% of hands:

On a 3-flush board showing A-K-Q, there are merely 3 hands (JTs, T9s, and 65s) that complete flushes for the player.

Conversely, if the board displays something like 8-7-4 all in one suit, all of a sudden the suited 'flushing' combinations within the UTG player’s range become accessible.

They would then have 20 flush combinations in total (comprising 8 non-nut flush and 12 nut flush combinations).

• Accordingly, it's vital to consider poker probability/combinatorics for flushes that accurately reflect a player’s range of suited holdings.

Note: On a board with 4 flush cards, it is necessary to factor in both suited hole cards and unsuited combinations. These hands should include one card of the flush when calculating combinations.

When discussing the combinatorics or possibilities for straights in poker, the dynamics become a bit more complex. There are generally numerous straight possibilities a player might have, which depend on the structure of the board.

The primary factor that determines how many combinations of straights a player can have in their range relates to the following:

Specifically, it hinges on the number of unsuited hand combinations at a given value that can form a straight .

• For instance, with a board of 9-8-6, T-7 would create the nut straight, yielding a total of 16 combinations.

However, if the player only possesses T7 suited and lacks T7 offsuit in their range, their nut straight combinations will diminish significantly by 75% – from 16 down to just 4!

FAQ Section

A: In Texas Hold’em, there are 1,326 distinct possible hole card combinations that can be dealt.

Q: How many combinations of paired hands exist in poker?

A: Each pocket pair has 6 combinations, which means there are a total of 78 combinations of pocket pairs across 13 different card values. Thus, the likelihood of being dealt a pocket pair is 78/1326 = 5.9% (which is about once every 17 hands).

Q: How many combinations of unpaired hands are there in poker?

A: Each type of unpaired hand has 16 combinations (4 suited and 12 unsuited). This results in 288 combinations of suited unpaired hands and 864 unsuited unpaired hand combinations.

A: For unpaired hands, simply take the number of available cards of one value and multiply it by the number of available cards of the other value.

Q: How do I calculate combinatorics?

For example, there are 16 combinations for AQ. To arrive at this figure, multiply (4 x Aces) with (4 x Queens). If there's an Ace visible on the board, only 3 Aces remain available. Thus, the calculation becomes (3 x Aces) times (4 x Queens), leading to 12 possible hole card combinations of AQ.

Q: What is the most important tip for counting combinations?

A: When considering combinatorics, it's crucial to avoid double counting poker hands. For instance, 8-7 of hearts is equivalent to 7-8 of hearts. Therefore, when performing specific combinatorial calculations, you need to divide your result by 2 to account for this equivalency.

Poker Combinatorics – Key Takeaways

Hopefully, this article has provided you with a more accessible combinatorics system applicable to calculating different hand combinations during gameplay.

Leverage this knowledge to enhance your skills in the following areas:

Learning how to estimate the likelihood of your opponents holding specific poker hands within their range.

• How poker ranges work
• Understanding how to maintain the ideal ratio between value and bluff combinations across various streets, and much more!
• Until we meet again, best of luck on the felt!

Matthew Cluff is a dedicated poker player, focusing on 6-Max No Limit Hold’em formats. He also occasionally contributes online poker content to various platforms.