72 off-suit is mathematically the worst starting hand you can have in Texas Holdem. In fact, many home or cash games on TV have a bonus for winning with this hand to induce action. Do you shuffle after EVERY Texas Holdem hand? It's called the Shuffle and Cut - and it's done after every hand. One deck per hand in Texas Hold'em and all other player vs player (poker-inspired, player vs casino table games may be different, I don't know anything about them) poker games of which I am aware. Charts ranking the 169 different starting hands in Texas hold'em can be very intriguing, but it's important to remember they are of limited value. The total of 169 combinations represents a. A number of poker experts have written on the subject of good poker starting hands for Texas Holdem. Beginners should read the best poker writers to further their study of poker hand strength. David Sklansky and Mason Malmuth of 2+2 fame designed an influential set of Texas Holdem starting hand charts in 1999. In poker terms, the river is the name for the fifth card dealt, face-up on the board. In total, there are 2,598,960 possible poker hands with 52 cards. The odds of getting four of a kind in Texas Hold ‘Em is 4164 to 1.
Is poker a game of skill or chance? This question has been discussed and
argued in many places and is the center of the arguments for and against
legalizing Texas holdem and other forms of poker in many places, including
online.
The answer to this question boils down to the mathematics behind the game. If
the math shows one player can win more often than another based on the
mathematical and statistical truths about Texas holdem then the game is one of
skill.
Let’s look at a few facts before moving on.
- Fact 1
Texas holdem is played with a deck of 52 playing cards, consisting of
the same four suits, and 13 ranks in every deck. You know each deck has an
ace of spades, and ace of hearts, an ace of clubs, and an ace of diamonds.
The same is true for kings, queens, and all of the ranks down through twos. - Fact 2
Over a long period of time each player will play from each position at
the table an equal number of times. In other words, each player will play in
the small blind, the big blind, under the gun, on the button, etc. an equal
number of times as other players. If you take two individual players it
might not be 100% the same, but it’ll be close. When you take thousands of
players and average their times played in each position mathematically they
each play the different positions an equal number of times. - Fact 3
The rules in each game are the same for every player at the table.
- Fact 4
The player that starts the hand with a better two card starting hand
wins the hand more often than the player with a worse hand. This has been
proven by computer simulations that run millions of hands and consider every
possible outcome.
Why Is This Important?
The reason all of this is important to Texas holdem players is that you can
use all of this math to help you win.
Though there are thousands of possibilities on every hand of Texas holdem,
you can use the fact that everything is based on a set of 52 cards to predict
outcomes and possibilities at every stage for every hand.
If you start the hand with two aces as your hole cards, you know that the
remaining 50 cards in the deck only have two aces. The remaining 48 cards
consist of four of each rank below the aces. At the beginning of the hand you
don’t know where any of the other cards are located, but as the hand progresses
you learn where some of them are located.
Continuing with the example, if the flop has an ace and two fours, you hold a
full house. You also know the only hand at this time that can beat you is four
fours. Because two fours are on the flop, the number of times a single opponent
has the other two fours is 1 in 1,326 hands. This is such a small percentage of
the time that you always play the full house in this example as if it’s the best
hand.
How do we know the number of times the opponent has the other two fours?
Because two fours are on the flop, let’s say the four of hearts and the four
of diamonds, so you know that your opponent has to have the four of clubs and
the four of spades. The chances of the first card in their hand being one of
these two cards are two out of 52. If they get one of them as the first card
that leaves the single other card they need out of 51 unseen cards, or one out
of 51.
You multiply two over 52 times one over 51 and this gives us the 1 out of
1,326 hands.
Basic Texas Holdem Math
Some of the math we discuss on this page can be complicated and the truth is
some players won’t be able to use it all. But that doesn’t mean they can’t be
winning Texas holdem players. The math covered in this section forms the
building blocks for the advanced math covered lower on the page.
Every Texas holdem player can use the basic math included in this section,
and if you aren’t using it yet you need to start right away.
Starting Hands
At the most basic level of Texas holdem everything starts with your starting
hand. As we mentioned above, mathematically the player who stars the hand with
the better starting hand wins more than the player with the inferior hand.
This means the first math lesson you need to learn and start using is to play
better starting hand on average than your opponents. While this can get
complicated, especially in games with many multi way pots, you still need to
learn how to play better starting hands.
If you take nothing else from this page, if you simply tighten up your
starting hand selection it’ll immediately improve your results.
Position
It’s difficult to directly relate position to mathematics, but the main thin
to know is the later your position, the better your chances to play in a
positive expectation situation. We’ll discuss expectation in a later section,
but it’s important to understand that having position on an opponent is a strong
advantage that equates to a mathematical advantage over the long run.
Outs
One of the most important skills Texas holdem players need to develop is the
ability to determine the number of outs, or cards remaining in the deck that can
complete the hand they’re drawing to. You use this information to determine your
chances of winning the hand as well as to determine the pot odds. Pot odds are
discussed in the next section, but they show you whether or not a call is
profitable in the long run when an opponent makes a bet.
We can determine how many outs you have because we know what’s in the deck
and what we need to improve our hand. If you have a king, queen, jack, and 10
after the turn you know any of the four aces or four nines complete your
straight.
This means you have eight outs. You’ve seen six cards, so the deck has 46
cards remaining in it. Don’t make the mistake of thinking about the cards that
have been folded or your opponent holds. You haven’t seen these cards so any
unseen card is still considered a possible river card.
In other words, on average, if you play this situation 46 times you’re going
to complete your straight eight times and not complete it 38 times.
You should always consider how many outs you have in every situation while
playing. B knowing your outs you have another piece of information that can help
you make profitable decisions throughout the hand.
Pot Odds
The next question many players ask after they learn how to determine their
out sis how they can use this information to make more money at the table. This
is where pot odds come into play.
Pot odds are simply a ratio or comparison between the money in the pot and
the chances you have of completing your hand. You use this ratio to determine if
a call or fold is the best play based on the information you currently have.
If you consider the example in the last section concerning the straight draw,
you know that the deck holds eight cards that complete your straight and 38
cards that don’t. This creates a ratio of 38 to 8, which reduces to 4.75 to 1.
You reduce by dividing 38 by 8.
The way you use this ratio is by comparing it to the amount of money in the
pot and how much you have to put into the pot. If the pot odds are in your favor
it’s profitable to call and if not you should fold.
If the pot has $100 in it and you have to make a $10 call the pot is offering
10 to 1 odds. You determine this the same way as above, by dividing $100 by $10.
If you’re in the situation described above of drawing to a straight on the
river you can see that a call is correct because the pot is offering 10 to 1 and
you have a 4.75 to 1 chance of winning.
On the other hand of the pot has $100 in it and you have to put $40 in to see
the river the pot is only offering 2.5 to 1 odds and your chances of hitting
your straight are still 4.75 to 1 so you should fold.
Pot odds can get complicated, especially when you start considering how they
work when you’re determining the correct play with both the turn and river to
come.
Fortunately charts are available to quickly check the odds of hitting your
hand based on how many outs you have. We’ve included one next so all you have to
do is determine your outs and compute the odds the pot is offering. Then compare
the two to see if it’s profitable to call or fold.
| Number of Outs | Turn & River Combined | River Only |
|---|---|---|
| 1 | 22.26 to 1 | 45 to 1 |
| 2 | 10.9 to 1 | 22 to 1 |
| 3 | 7 to 1 | 14.33 to 1 |
| 4 | 5.06 to 1 | 10.5 to 1 |
| 5 | 3.93 to 1 | 8.2 to 1 |
| 6 | 3.15 to 1 | 6.67 to 1 |
| 7 | 2.6 to 1 | 5.57 to 1 |
| 8 | 2.17 to 1 | 4.75 to 1 |
| 9 | 1.86 to 1 | 4.11 to 1 |
| 10 | 1.6 to 1 | 3.6 to 1 |
| 11 | 1.4 to 1 | 3.18 to 1 |
| 12 | 1.22 to 1 | 2.83 to 1 |
| 13 | 1.08 to 1 | 2.54 to 1 |
| 14 | 0.95 to 1 | 2.29 to 1 |
| 15 | 0.85 to 1 | 2.07 to 1 |
| 16 | 0.75 to 1 | 1.88 to 1 |
| 17 | 0.67 to 1 | 1.71 to 1 |
| 18 | 0.6 to 1 | 1.56 to 1 |
| 19 | 0.54 to 1 | 1.42 to 1 |
| 20 | 0.48 to 1 | 1.3 to 1 |
Expand Shrink
When you’re determining your pot odds for the turn and river you determine
them on the turn and then if you don’t hit your draw you determine them again on
the river. This often happens, especially in limit Texas holdem. But if an
opponent moves all in on the turn you simply use the turn and river combined
odds in your decision.
Advanced Texas Holdem Math
Many beginning Texas holdem players look at a discussion about expectation
and instantly decide it’s too hard and ignore it. When they do this they
severely hurt their long term chances at being a profitable player.
We’ve broken down how to look at situations while playing poker in a simple
manner that almost any player can use below. Do yourself a favor and go into
this with an open mind. Once you understand it at a simple level you can learn
more as you gain experience. You may be surprised at just how easy it gets to
determine positive and negative expectation with a little practice.
Expectation
Expectation is what the average outcome will be if you play the same
situation hundreds or thousands of times. Once you determine the expectation you
know if a situation offers positive or negative results on average.
Your goal as a Texas holdem player is to play in as many positive expectation
situations as possible and avoid as many negative expectation situations as
possible.
You need to understand that expectation is something that can be applied to
almost any situation in poker, but it’s also subjective in many areas.
- If you play at a table where every opponent is better than you in the long
run you’re going to lose money. This is a negative expectation situation. - If you play at a table where every opponent is a worse
player than you it’s a positive expectation situation because you’re going to
win in the long run.
The problem is determining whether a situation is positive or negative
expectation when you sit down at a table with some players who are better than
you and some who are worse.
You can find many situations where it’s easier to determine expectation
mathematically, and we’ll teach you how to do this now. While this may seem
overly complicated at first, especially to do at the table while playing, you
don’t need to know exactly how negative or positive a situation is, you only
need to know if it’s positive or negative.
Once you determine if a situation is positive expectation or negative
expectation you simply remember the next time you’re in a similar situation.
Once you start determining expectation you’ll find that you learn mist
situations quickly and only have to think through an occasional situation at the
table.
The best way to see how to determine expectation is by running through a
couple examples.
Example 1
You’re facing a bet after the turn and you have four to a flush.
The pot had $400 in it and your opponent bet $100. You’re certain that if you
miss your flush draw you’ll lose and when you hit your flush draw you’ll win.
In order to see the river you have to call the $100 bet. When you lose you
lose $100, and when you win you get back $600. You get your $100 back plus the
$400 that was in the pot plus the $100 bet your opponent made.
Many players claim that part of the money already in the pot is theirs, but
once you put money into the pot it isn’t yours. The only way to get it back is
to win the pot. So you can’t consider it in any other way when determining
expectation.
The way to see if it’s positive or negative to call is to determine what will
happen on average if you play the same situation many times. Most players find
it easiest to determine by pretending to play the hand 100 times.
In this example you’re going to hit your flush 9 out of 46 times. This means
19.56% of the time you’re going to win and 80.44% of the time you’re going to
lose. To make this simple we’ll round these numbers off to 20% and 80%.
Percentage Of Starting Hands In Texas Holdem
If you have to put $100 in the pot 100 times your total investment is
$10,000. The 80 times you lose you get nothing back. The 20 times you win you
get $600. 20 times $600 is $12,000. When you take the $12,000 you win and
subtract the $10,000 you lose when you play the situation 100 times, you see
that you win $2,000 overall.
To determine how much you win on average per hand simply divide the $2,000 by
100 to get a positive expectation of $20 per hand. This means that every time
you’re in this situation you’ll win on average $20.
The truth is you may win a little more because we’re ignoring the river.
Because you know you can’t win if you miss your flush, you always need to fold on
the river when you miss your draw. Every once in a while you may be able to
extract a small bet from your opponent on the river when you hit your flush,
increasing your average expectation. Sometimes it’s even correct for your
opponent to call on the river in this situation. See the next example to see
why.
Example 2
Let’s say you’re playing the same hand as above but you have a
straight and your opponent appears to be drawing to a flush. You’re on the
river, the pot has $600 in it, and the board has the third suited card hit on the
river.
If your opponent was drawing to the flush, they completed it and you’re going
to lose the hand. In this situation your opponent bets $20.
In this situation you clearly have to call.
The reason you have to call is because you can’t know for certain your
opponent was drawing to the flush. They may be bluffing or have two pair or any
other number of hands that aren’t as good as your straight.
Let’s look at the math behind this decision.
If you play the situation 100 times your total investment is $20 times 100,
or $2,000.
When you win you get $640, consisting of the original $600 pot, your
opponent’s $20 bet, and your $20 call. If you win three hands you get back
$1,920 for a loss of $80, or 80 cents per hand.
If you win at least four times you’re in a positive expectation situation.
Four wins nets $2,560 for an overall win of $560, or $5.60 per hand.
How Many Different Starting Hands In Texas Holdem
What this means is if your opponent is bluffing or has a weaker hand just
four times out of 100 or more, calling is a positive expectation situation. Four
times out of 100 is only 4%. You’ll win at least 4% of the time in this
situation.
The numbers get closer the more your opponent bets on the river, and the
closer the numbers get the more you’re going to need to use what you know about
your opponent to determine if a situation is positive or not.
Start looking at every decision you make at the Texas holdem tables in terms
of positive and negative expectation.It’s hard at first, but the more you
practice the better you’ll get at predicting if a situation offers positive
expectation.
Summary
Texas holdem math is often the only thing that separates winning and losing
players. Take the time to learn the basics now so you can improve your game in
every way possible as you gain experience. This guide is the perfect place to
start for players of every experience level.
For a great training video on poker combinatorics, check out this poker combos video.
'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.
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
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).
Total Possible Hands In Texas Holdem
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
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 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.
Total Number Of Starting Hands In Texas Holdemhold Em
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.
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%)
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: 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.
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…
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.
Total Number Of Starting Hands In Texas Holdemem
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)
Number Of Starting Hands In Texas Holdem
If you’re more interested in finding out more about combinations in poker only, here are a few interesting reads:
Total Number Of Starting Hands In Texas Holdem Rules
Go back to the awesome Texas Hold'em Strategy.
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