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For a great training video on poker combinatorics, check out this poker combos video.
Texas Holdem Poker Two Card Hand Frequencies. Both before and after the flop in any limit Holdem game, it is helpful to be able to make a reasonably accurate estimate of the types and numbers of playable hands that you are likely to be competing against. Apr 14, 2009 Since a definitive guide on every hand and how and when to play it in every situation would take more words than a novel, this article will touch on the major points of basic pre-flop hands with broad strokes. Texas Holdem Starting Hands. Watch the video below for some immediate help picking the right Texas Hold'em Starting Hands. Jul 27, 2018 Minimum defense frequencies are your shield. Minimum defense frequency (MDF). Texas Hold’em Poker Hand Example. Let’s use MDF and pot odds to analyze the following hand history from both players’ perspectives. 100NL, 100BB Effective Stacks. Preflop Folds to CO who opens to 2.25BB, 2 folds, BB calls.
'Combinatorics' is a big word for something that isn’t all that difficult to understand. In this article, I will go through the basics of working out hand combinations or 'combos' in poker and give a few examples to help show you why it is useful.
- As a baseline we'd recommend raising three times the big blind with hand like like 22+/AT+/KJ+/89s-JQs/A2s-A5s. This is 15.5% of hands. If we have tight players on our left we can start to raise.
- This is a discussion on Royal flush frequencies within the online poker forums, in the Cash Games section; I play on a very small site that has 200 people seated at any given time on average.
- Going beyond Texas hold'em starting hand charts. Choosing your starting hands wisely can make the difference between winning and losing in poker. This is especially true when you can't depend on your poker skills to help you out after the flop in more difficult situations (yet).
- Since a definitive guide on every hand and how and when to play it in every situation would take more words than a novel, this article will touch on the major points of basic pre-flop hands with broad strokes. Texas Holdem Starting Hands. Watch the video below for some immediate help picking the right Texas Hold'em Starting Hands.
Oh, and as you’ve probably noticed, 'combinatorics', 'hand combinations' and 'combos' refer to the same thing in poker. Don’t get confused if I use them interchangeably, which I probably will.
What is poker combinatorics?
Poker combinatorics involves working out how many different combinations of a hand exists in a certain situation.
For example:
- How many ways can you be dealt AK?
- How many ways can you be dealt 66?
- How combinations of T9 are there on a flop of T32?
- How many straight draw combinations are there on a flop of AT7?
Using combinatorics, you will be able to quickly work these numbers out and use them to help you make better decisions based on the probability of certain hands showing up.
Poker starting hand combinations basics.
- Any two (e.g. AK or T5) = 16 combinations
- Pairs (e.g. AA or TT) = 6 combinations
If you were take a hand like AK and write down all the possible ways you could be dealt this hand from a deck of cards (e.g. A K, A K, A K etc.), you would find that there are 16 possible combinations.
- See all 16 AK hand combinations:
Similarly, if you wrote down all the possible combinations of a pocket pair like JJ (e.g. JJ, JJ, JJ etc.), you would find that there are just 6 possible combinations.
- See all 6 JJ pocket pair hand combinations:
So as you can see from these basic starting hand combinations in poker, you’re almost 3 times as likely to be dealt a non-paired hand like AK than a paired hand. That’s pretty interesting in itself, but you can do a lot more than this…
Note: two extra starting hand combinations.
As mentioned above, there are 16 combinations of any two non-paired cards. Therefore, this includes the suited and non-suited combinations.
Here are 2 extra stats that give you the total combinations of any two suited and any two unsuited cards specifically.
- Any two (e.g. AK or 67 suited or unsuited) = 16 combinations
- Any two suited (AKs) = 4 combinations
- Any two unsuited (AKo) = 12 combinations
- Pairs (e.g. AA or TT) = 6 combinations
You won’t use these extra starting hand combinations nearly as much as the first two, but I thought I would include them here for your interest anyway.
It’s easy to work out how there are only 4 suited combinations of any two cards, as there are only 4 suits in the deck. If you then take these 4 suited hands away from the total of 16 'any two' hand combinations (which include both the suited and unsuited hands), you are left with the 12 unsuited hand combinations. Easy.
Fact: There are 1,326 combinations of starting hands in Texas Hold’em in total.
Working out hand combinations using 'known' cards.
Let’s say we hold KQ on a flop of KT4 (suits do not matter). How many possible combinations of AK and TT are out there that our opponent could hold?
Unpaired hands (e.g. AK).
How to work out the total number of hand combinations for an unpaired hand like AK, JT, or Q3.
Method: Multiply the numbers of available cards for each of the two cards.
Word equation: (1st card available cards) x (2nd card available cards) = total combinations
Word equation: (1st card available cards) x (2nd card available cards) = total combinations
Example.
If we hold KQ on a KT4 flop, how many possible combinations of AK are there?
There are 4 Aces and 2 Kings (4 minus the 1 on the flop and minus the 1 in our hand) available in the deck.
C = 8, so there are 8 possible combinations of AK if we hold KQ on a flop of KT4.
Paired hands (e.g. TT).
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How to work out the total number of hand combinations for an paired hand like AA, JJ, or 44.
Method: Multiply the number of available cards by the number of available cards minus 1, then divide by two.
Word equation: [(available cards) x (available cards - 1)] / 2 = total combinations
Word equation: [(available cards) x (available cards - 1)] / 2 = total combinations
Example.
How many combinations of TT are there on a KT4 flop?
Well, on a flop of KT4 here are 3 Tens left in the deck, so…
C = 3, which means there are 3 possible combinations Contoh rancangan pengajaran slot prasekolah. of TT.
Thoughts on working out hand combinations.
Working out the number of possible combinations of unpaired hands is easy enough; just multiply the two numbers of available cards.
Working out the combinations for paired hands looks awkward at first, but it’s not that tricky when you actually try it out. Just find the number of available cards, take 1 away from that number, multiply those two numbers together then half it.
Note: You’ll also notice that this method works for working out the preflop starting hand combinations mentioned earlier on. For example, if you’re working out the number of AK combinations as a starting hand, there are 4 Aces and 4 Kings available, so 4 x 4 = 16 AK combinations.
Why is combinatorics useful?
Because by working out hand combinations, you can find out more useful information about a player’s range.
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For example, let’s say that an opponents 3betting range is roughly 2%. This means that they are only ever 3betting AA, KK and AK. That’s a very tight range indeed.
Now, just looking at this range of hands you might think that whenever this player 3bets, they are more likely to have a big pocket pair. After all, both AA and KK are in his range, compared to the single unpaired hand of AK. So without considering combinatorics for this 2% range, you might think that the probability break-up of each hand looks like this:
- AA = 33%
- KK = 33%
- AK = 33%
…with the two big pairs making up the majority of this 2% 3betting range (roughly 66% in total).
However, let’s look at these hands by comparing the total combinations for each hand:
- AA = 6 combinations (21.5%)
- KK = 6 combinations (21.5%)
- AK = 16 combinations (57%)
There are more AK hands in a range of [AA, KK, AK] than there are AA and KK hands combined.
So out of 28 possible combinations made up from AA, KK and AK, 16 of them come from AK. This means that when our opponent 3bets, the majority of the time he is holding AK and not a big pocket pair.
Now obviously if you’re holding a hand like 75o this is hardly comforting. However, the point is that it’s useful to realise that the probabilities of certain types of hands in a range will vary. Just because a player either has AA or AK, it doesn’t mean that they’re both equally probable holdings - they will actually be holding AK more often than not.
Analogy: No deposit casinos. If a fruit bowl contains 100 oranges, 1 apple, 1 pear and 1 grape, there is a decent range of fruit (the 'hands'). However, the the fruits are heavily weighted toward oranges, so there is a greater chance of randomly selecting an orange from the bowl than any of the 3 other possible fruits ('AK' in the example above).
This same method applies when you’re trying to work out the probabilities of a range of possible made hands on the flop by looking at the number of hand combinations. For example, if your opponent could have either a straight draw or a set, which of the two is more likely?
Poker combinatorics example hand.
You have 66 on a board of A J 6 8 2. The pot is $12 and you bet $10. Your opponent moves all in for $60, which means you have to call $50 to win a pot of $82.
You are confident that your opponent either has a set or two pair with an Ace (i.e. AJ, A8, A6 or A2). Don’t worry about how you know this or why you’re in this situation, you just are.
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According to pot odds, you need to have at least a 38% chance of having the best hand to call. You can now use combinatorics / hand combinations here to help you decide whether or not to call.
Poker combinatorics example hand solution.
First of all, let’s split our opponent’s hands in to hands you beat and hands you don’t beat, working out the number of hand combinations for each.
Adding them all up…
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Seeing as you have the best hand 79% of the time (or 79% 'equity') and the pot odds indicate that you only need to have the best hand 38% of the time, it makes it +EV to call.
So whereas you might have initially thought that the number of hands we beat compared to the number of hands we didn’t beat was close to 50/50 (making it likely -EV to call), after looking at the hand combinations we can see that it is actually much closer to 80/20, making calling a profitable play.
Being able to assign a range to your opponent is good, but understanding the different likelihoods of the hands within that range is better.
Poker combinatorics conclusion.
Working out hand combinations in poker is simple:
- Unpaired hands: Multiply the number of available cards. (e.g. AK on an AT2 flop = [3 x 4] = 12 AK combinations).
- Paired hands: Find the number of available cards. Take 1 away from that number, multiply those two numbers together and divide by 2. (e.g. TT on a AT2 flop = [3 x 2] / 2 = 3 TT combinations).
By working out hand combinations you can gain a much better understanding about opponent’s hand ranges. If you only ever deal in ranges and ignore hand combinations, you are missing out on useful information.
It’s unrealistic to think that you’re going to work out all these hand combinations on the fly whilst you’re sat at the table. However, a lot of value comes from simply familiarising yourself with the varying probabilities of different types of hands for future reference.
For example, after a while you’ll start to realise that straight draws are a lot more common than you think, and that flush draws are far less common than you think. Insights like these will help you when you’re faced with similar decisions in the future.
The next time you’re doing some post session analysis, spend some time thinking about combinatorics and noting down what you find.
Poker combinatorics further reading.
Hand combinations in poker all stem from statistics. So if you’re interested in finding out more about the math side of things, here are a few links that I found helpful:
- Combinations video - Youtube (all the stuff on this channel is awesome)
If you’re more interested in finding out more about combinations in poker only, here are a few interesting reads:
Also, I’d highly recommend you check out the Mathematics of NL Hold’emEp. 2 and Ep. 6 by WiltOnTilt at DeucesCracked for some great video tutorials (and examples) on using hand combinations in poker.
Go back to the awesome Texas Hold'em Strategy.
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Once you familiarize yourself with basic Texas Hold ‘em play, hand rankings and dealing, you should learn about the odds and probabilities associated with different aspects of the game. While it may be impossible to calculate precise odds mid-hand, particularly given you’ll never see your opponents’ cards until all decisions have been made, knowing when to play (and when not to play) different hands typically requires more than just good instinct.
In Texas Hold ‘em, winners are often decided before the community cards are even laid out, so the chances of being dealt a strong starting hand is something many players think.
Calculating Texas Hold ‘em Odds
To understand the odds of drawing any particular starting hand, you’ll have to consider all of the factors that go into the deal of a hand. Considering a shuffle to completely randomize the order of the cards, the biggest factors that will affect a draw are the number of players at the table and your own position relevant to the dealer. Also consider that there are four cards of any given value in each deck and there are 169 different hole card combinations. That being said, there’s a roughly one in 1326 chance that players will be dealt any specific hand, so it’s important to know what kind of odds you should keep in mind whenever you step to the poker table.
Pre Flop
You have a 55:1 chance of being dealt a pair of Jacks or higher.
Your Texas Hold’em experience will begin with your hole cards, and while you cannot change the cards that are dealt your way, it’s good to know what to expect. While World Series of Poker reruns may make it seem like someone at the table will be dealt pocket kings every hand, starting with a monster hand is rarer than you may imagine. For more on Pre Flop strategy see our guide here.
Quick Odds
While an extensive list of the probabilities for preflop hands could fill a book, here are some quick odds to consider when you’re peeking at your cards for the first time.
- Being dealt a pocket pair: 16:1
- Being dealt suited cards: 3.2:1
- Being dealt suited connectors: 24:1
- Being dealt AK: 82:1
- Being dealt two cards of jack or higher: 10:1
- Being dealt AA, KK, QQ or JJ: 55:1
The Flop
The flop is a game changer in every sense of the words. With more cards in your hand, players will get their first real taste of what the final outcome will look like. That being said, players will want to think about the odds for how their hand could turn out before they start placing or calling bets. For more on Post Flop strategy see our guide here.
Quick Odds
Though the presented odds may be mathematically accurate, they do present a best case scenario that you won’t always find yourself in at the table. These calculations don’t take into account the probabilities of the card or cards you are looking for turning up as burn cards or in your opponents’ hands, so keep that in mind when considering your next move.
- Hitting a pair on the flop: 2:1
- Pocket pair improving to three of a kind on the flop: 7:1
- Flop being three of a kind: 424:1
- Odds of flopping a flush with suited hole cards: 118:1
The odds of flopping a straight can be a bit harder to pin down. Depending on your starting hand, there are any number of card combinations that could lead to a flopped straight. Those hands that have the more flexibility when it comes to straight draws, particularly middling connectors (those between 5 and 9), have the strongest chances of hitting on the flop because there are more cards that could allow for their straight to hit. The wider the gap between your starting cards, the worse your chances are.
The Turn and River
Naturally, as the hand progresses and fewer cards remain to be dealt, the odds of making any particular hand become slimmer and slimmer. That being said, you’ll want to look at the odds for a hand from beginning to end as well as those from stage to stage. Poker starting hand chart.
Quick Odds
Your odds change as each new card is dealt.
Over the course of a full hand, these scenarios could play out
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- One pair on flop improving to two pair or a set by the river: 4:1
- Catching an ace on the turn or river: 7:1
- Open ended straight hitting by the river: 2:1
- Gutshot straight draw hit by the river: 5:1
- Pocket pair improving to set by river: 10:1
- Going runner runner for a flush: 23:1
- Going runner runner for a straight: 67:1
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Pot and Implied Odds
In addition to the odds that your cards will pair or better, experienced players know that pot odds, that is, the ratio of the size of the pot to the cost of a call, play an important role in proper Hold’em strategy. With all of the risks that are inherently involved in games like poker, players won’t get far if they don’t decide to gamble every so often – and pot odds allow you to make smarter decision when it comes time to place your money on the line. Pot odds take into consideration the likelihood of making a hand to establish a decent risk vs. reward ratio.
Another important statistic for players to keep in mind is ‘implied odds,’ which factor in the likely size and frequency of any betting that may come in later rounds of play. These come into play post flop and will be a factor until the river is dealt, and unlike most other Texas hold ’em odds, implied odds will be entirely based on your read on your opponents’ betting habits and aggression.
To calculate effective pot odds, consider your position in the hand and relative strength, then make a note of the amount of money currently in the pot. If you’re in a position to call, think of how much you stand to win should your hand come out ahead, as establishing a solid return on investment can help turn a questionable call into a big win. If no one has bet, then pot odds can help you deduce what size bet you will want to make in order to shift the statistical probabilities in your favor. With a sound bet, you could either make the continued investment not worth the potential gain or place a value wager designed to elicit a call from any remaining opponents. You will have to decide whether or not your chances of winning are worth risking the amount you could lose if you are wrong – a fine line for some players, but one that must be crossed often. For more on Texas Hold’em pot size and strategy see our guide here.