Poker Combinations
One of the more math-intensive strategy topics in poker is figuring out the chances of being dealt different hands. The type of poker you are playing determines whether you use five or seven cards to find the value of your hand, and this by itself can drastically change the chances of making particular poker hands.
Along these lines, we use a mathematical idea called a combination to figure out the probability of different poker hands in different situations.
A combination is when you have a certain number of items in a set, and you want to know how many different ways you can choose a specific number of those items.
In mathematics, this is written in the format "n choose r" meaning there are n options to choose from, and we want to pick r of them. The formula needed to evaluate a statement of this form is the following:
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We'll be using this formula in our discussion of poker probabilities relating to hand combinations. However, if you aren't comfortable writing out the math on your own, you can still use a calculator to find the value of combinations. Typing "52 choose 4" into Google, for example, will tell you how many different ways you can be dealt a four-card starting hand in Omaha.
Five Card Poker Hands
In games like five card stud and five card draw, we need to be able to figure out the chance of getting a certain hand by just being dealt five random cards.
The following chart shows the chance of being dealt these hands right off the bat by listing the number of combinations possible for each poker hand rank.
Hand | Combinations | Chance |
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 Pair | 123,552 | 4.75% |
One Pair | 1,098,240 | 42.3% |
High Card | 1,302,540 | 50.1% |
To find the total number of combinations of a five card hand, you'll do 52 choose 5 as follows:
You divide the number of combinations of a single hand by this value to get the chance of being dealt that hand with five random cards from a deck. This will tell you how likely you are to be dealt a specific hand which is an exceptionally important piece of information in games that use five cards.
Seven Card Poker Hands
Playing with seven card poker hands is very interesting, and this is what happens with holdem and Omaha (2 hole cards + 5 community cards), the two most popular games played today.
In these games, poker combinations and math are less about knowing the chances of being dealt a specific hand but about using these ideas to figure out the probability of your opponents having certain holdings.
Unique and Non-unique Starting Hands in Omaha
If you have the hand A♠ A♥ K♠ K♥ in Omaha, that's known as a distinct or unique combination of cards because there's only one way you can be dealt that hand. However, if you just say that you were dealt AAKK double-suited, then there could be several different ways of being dealt that particular hand, and you would have to figure out how many different unique combinations there were by using two combinations of suits like so:
- A♠ A♥ K♠ K♥
- A♠ A♣ K♠ K♣
- A♠ A♦ K♠ K♦
- A♥ A♣ K♥ K♣
- A♥ A♦ K♥ K♦
- A♣ A♦ K♣ K♦
Each of these six examples are unique combinations because there are no other ways to be dealt the hand. The reason that unique combinations are so important is that they help you to find the probability of actually being dealt those hands as you're about to see.
Finding the Probability of Texas Holdem Hands Using Combinations
In Texas holdem, you have the best opportunities for using the poker combinations probability to increase your chances of winning because it's how you evaluate what your opponents might be holding. You can also use it to determine how often you will be dealt certain hands.
Hand reading in Texas Holdem requires you to know the relative chances of your opponent having different hands in different situations.
For example, how many ways can you be dealt AA? Look at the possible combinations of suits to figure this out: spade-club, spade-heart, spade-diamond, club-heart, club-diamond and heart-diamond. That's a total of six. Remember that A♠ A♥ is the same thing as A♥ A♠, so they aren't counted as two different hands.
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To figure out how many total ways you can be dealt a starting hand, you'll calculate the combination 52 choose 2 to find the total number of possible starting hands, and then you do the division to find the chances of being dealt AA:
So there's a 0.45 percent chance of being dealt AA, and this process can be repeated for any pocket pair since they will all have an equal chance of being dealt.
There are 16 ways to be dealt a non-pair hand, and you can use the same process as above to find that there's a 1.21 percent chance of being dealt a specific one.