If you have ever been in a situation where you face a big bet and can't decide whether to call or fold with your weak hand, then you must know the feeling of your mind going blank in the most important moment.
Poker is undoubtedly entertaining, but what makes the casino game even spicier is the knowledge behind the mechanics of the card game. This is when poker combinatorics come into play.
Poker combinatorics is exactly what you need to know when estimating how many combos you may hold by the end of a showdown and whether they are worth the risk.
Without further ado, Natural8 is here to teach you the formulas to keep in mind when playing against fierce competitors.
What Is Poker Combinatorics?
Combinatorics evaluates the possible number of poker hand combinations in any given situation and of any given card. The term may sound like something difficult, but the truth is there is a set number of hand combinations, and they are fairly easy to remember.
Understanding combinatorics will answer questions that will help you make the right decision during the game. You will learn how to calculate the probability of getting AK, 66, or even a flush draw.
How Knowing Poker Hand Combinations Can Help You
If you're playing poker blindly, chances are you think it's a luck game, and you have no means to predict your future hand combination. That's very far from the truth.
To some extent, poker is based on maths. There are formulas you can apply during the game that will help with:
Understanding poker hand ranges
Understanding when to fold, raise, or call
Predicting how strong other players' hand
Understanding the game's mechanics better
By devoting some of your time to poker combinatorics, you will acquire valuable information that can be used against your opponents, especially when money is at stake.
How Many Poker Combos Are There?
All figures added up together equal 1,326 different poker hand combinations. Let's break the number down. The charts below may give you a back-to-school vibe, but these aren’t the numbers they tell you about at school, so pay close attention!
When researching hand combinations, you will often find a chart featuring 169 poker hands.
Here is the exact same grid but with the indication of the number of the corresponding poker hand combinations.
What we have is 1,326 poker combos that are divided into these 3 categories:
Paired hands - 78 combos
Suited unpaired hands - 312 combos
Unpaired unsuited hands - 936 combos
What do these poker hand combinations tell you? For example, you are 3 times more likely to get unpaired and unsuited hands (e.g., AK or 94) than a pocket pair -- and that's just the tip of the iceberg.
Calculating Remaining Hand Combinations
When you're training for a poker tournament, there is plenty of time to do your maths and count combos. However, during a live poker game, it may be challenging. But I have a solution for that, too.
To work out hand combos of your opponents, I will use real-life examples.
Suppose you are holding an unpaired hand of KQ on a flop of KT4. What are the chances that your opponent has a set or two pair with Ten?
Unpaired Hands
How many combos of unpaired hands can your opponent have? To find out, you will need a formula.
C = A1 x A2
C stands for total combinations. A1 stands for available cards for the first cards. A2 stands for available cards for the second card.
If you have KQ on a flop of KT4, how many possible combinations of AK can there be? If we have 4 aces and 2 remaining Kings (excluding the one in your hand and the one on the flop), the formula is
C = 4 x 2
C = 8
Hence, there are 8 possible combinations of AK in the given situation.
To master the formula, try it out for yourself with different community cards.
Paired Hands (Pocket Pair)
How many combinations of paired hands can there be? Here comes another formula to your rescue:
C = [A x (A - 1)] / 2
C stands for the number of possible combinations. A stand for the number of available cards.
Let's count combinations for TT, provided there is a flop of KT4. Here, we focus only on the card that we count combinations for, namely Ten. Since one Ten is a community card, there are 3 more remaining.
C = [3 x (3-1)] / 2
C = [3 x 2] / 2
C = 3
There are 3 possible combinations of TT.
You can simplify this formula by reducing the card's number by 1 and continuing the calculations.
Finding Out an Opponent's Range
Applying these simple formulas during your poker game will also reveal such information as opponents' range. How likely is the other poker player to hold a strong pocket pair or an unpaired hand? Poker combinatorics will also help you estimate the likelihood of other players having straight or flush draws.
If we take a 2% range of an opponent's 3betting, it's likely to be the AK-suited hand or powerful pocket pairs like AA and KK.
The probability of the other player having any of these is equally 33%. However, with our formulas applied, only one of the pairs appears more often than others -- AK. AK-suited hands have 16 combinations in total, unlike AA and KK, which have only 6 combos each.
Example of Counting Hand Combinations
Here is another example of how you can count hand combinations and pot odds. This is a personal example that happened to me during a game where poker strategy had little influence. But what did help me instill some confidence in my hand was counting combos. And I hope, one day, it will save you, too.
Suppose you hold a pocket pair of 66 while you see A, J, 8, 2, 6 on the poker table. Among all poker players, this is a 1vs1 game, where your opponent goes all in without thinking twice. You may instantly think that the other player has either a strong pair or a set.
To win the game, you need at least a 38% chance of having the best hand combination. The question is whether you should fold or call. The number of available cards is more than enough to count the chances of weak and strong hand combinations you can or cannot handle.
Let's divide hand combos into ones that you can beat:
| Beatable hand combinations | Unbeatable hand combinations |
|---|---|
| AJ | AA |
| A8 | JJ |
| A6 | 88 |
| A2 | - |
| 22 | - |
Now is the time to apply our formulas and see whether your Three-of-a-Kind has any advantage in this situation.
| Beatable hand combinations | Unbeatable hand combinations |
|---|---|
| AJ = 3 x 3 = 9 combos | AA = (3 x 2) / 2 = 3 combos |
| A8 = 3 x 3 = 9 combos | JJ = (3 x 2) / 2 = 3 combos |
| A6 = 3 x 1 = 3 combos | 88 = (3 x 2) / 2 = 3 combos |
| A2 = 3 x 3 = 9 combos | - |
| 22 = (3 x 2) / 2 = 3 combos | - |
If before you might have thought that the chances of winning the pot are 50/50, the truth is:
Beatable hand combos = 33 (79%)
Unbeatable hand combos = 9 (21%)
Since you only need to have the best hand combination 38% of the time, and you now have 79%, this is an indication that you can indeed make a call and snatch the prize, as it happened in my case.
Concluding Poker Card Combos
Understanding the opponent's range is one thing, but knowing your chances of winning can give you an idea of whether it is profitable to call or fold.
To count combinations of unpaired hands, you only need to multiply the number of available cards. To count combinations of paired hands, you need to deduct 1 from the number of available cards, multiply the remaining, and divide by 2.
Besides finding out possible combinations, you can also understand the opponent's range, which will give you useful insights during the game.
At first, it may be challenging or even time-consuming to calculate everything in your head. However, after some time, you start recognising which combinations are more common or less common.
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