• How Many Possible Hands In Texas Holdem Poker
• Top Hands In Texas Holdem
• How Many Cards In Hand Texas Holdem
• How Many Possible Hands In Texas Hold'em

In the last couple of years, as theoretical understanding of poker has galloped forward, an entire new vocabulary has emerged. I mean, when I was playing limit hold'em in San Jose 25 years ago, had you said, 'minimum defense frequency,' they'd have thought you were talking about the 49ers.

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. In Texas Hold’em poker there are 2,652 possible starting hands. The way that you first get all the possible starting hands is to take the number of cards (52) and multiply that by 51 times. Remember that the first 2 cards that can be dealt can be anything from the deck. Sometimes the best texas hold em hand is made by the five shared cards on their own. If they were 10-10-10-10-A and you had 9-9 your hole cards would not play as there is a four-of-a-kind of higher value already showing. All Texas Hold’em starting hands can be separated into two categories: “suited” and “offsuit”. Suited hands contain two cards of the same suit, like J♣9♣, A ♥ K ♥, K♠Q♠ and 9 ♦ 3 ♦. All other starting hands are in the offsuit category, like A♠8 ♦, 7♣5 ♥ and K ♥ 9 ♦.

One of the words that you hear most frequently now is 'combinations' or 'combos.' Once you accept the concept of an opponent (or yourself) having a range of hands, the next interesting question is, 'Well, how many hands are in that range?' The way you answer that is to figure out how many 'combinations' there are of the hands that make up the presumed range.

Counting Combos: Pocket Pairs

Let's try one. Suppose you are playing $1/$3 no-limit hold'em and before the flop, you raise to $12 in early position with a pair of jacks. A straightforward and uncreative player in late position reraises you to $40. It folds back around to you. Based on your knowledge of this player, you expect her to three-bet with only a pair of queens or better, and all of her ace-kings. So her three-betting range is Q-Q, K-K, A-A, and A-K. How do your jacks fare against her presumed range?

Well, there are six combos of every pocket pair. To determine that, we see that we can randomly pick any of the four queens in the deck, and now have three remaining queens with which to make a pair. So that's 4 x 3 = 12. However, if we pick the first, and then the , that's no different than picking the first and then the . So we must divide by two to get a total of six.

Another way of skinning the same cat: pick the and see that you can then pick the , , or to make a pair. That's three. Now pick the first, leaving just the and to pair. Two more. Finally, the has only the to pair it. One more. 3 + 2 + 1 = 6. Math is beautiful.

So, six combos for each pocket pair. For Q-Q / K-K / A-A, that's a total of 18 combos. So far so good.

Counting Combos: Non-Paired Hands

What about A-K combos? If we give the villain all of the ace-king combos, then she can make one by taking any of the four aces and crossing them with any of the four kings. 4 x 4 = 16 and that's the number of combos.

Of course, if she restricts herself to suited ace-kings, then pretty clearly there are just four of those — , , , .

Calculating Our Equity

We'd agreed that she would three-bet all of her ace-king combos plus queens, kings, or aces, so we conclude she has one of 34 possible hands: 18 pocket pairs and 16 ace-kings.

The 18 pocket pairs are 81-to-19 favorites against us, while we are a 57-to-43 favorite against the 16 ace-kings. To determine our equity against her, we weight each combo by its share of the range pie, compute our equity against that slice, and then sum them up.

For this example, we can compute our equity as follows:

1. Against the higher pairs, we have 0.18 (18%) equity. The higher pairs make up 18 / 34 of the villain's presumed range. So our equity for that piece = 0.18 x 18 / 34 = 0.10 or 10% (0.095 to be closer).
2. Against the ace-king combos, we have 0.57 (57%) equity. The ace-kings make up 16 / 34 of the villain's range. Our equity against that piece is 0.57 x 16 / 34 = 0.27 or 27%.
3. Summing the two, we get 0.10 + 0.27 = 0.37 or 37% equity against her presumed range.

The good news is that there are programs such as Pro Poker Tools and the like that let you ask questions such as 'How much equity does a pair of jacks have against a range of Q-Q / K-K / A-A / A-K?' But it's useful to know how those things are calculated.

Using Combos to Improve Decision-Making

What to do with that information is beyond the scope of this article, but as an example, if the villain were all in for her $40, we'd know exactly how to proceed.

Setting aside rake for the moment, there's $12 + $40 + $1 + $3 = $56 in the pot. It costs us another $28 to call. Conveniently enough, we're getting exactly 2-to-1 odds to call, so we must have at least 33.3% equity to call her shove. We have a hair above that (37%), so we shrug, slide in the extra $28, and ask the dealer to run out the board.

By the way, I had suggested that we ignore the rake for simplicity. Note that in this case once we take the rake into effect, this could turn into a fold. If you don't see that, subtract the rake ($5 or whatever) from the pot and redo the pot odds calculations, remembering that you still need to call the full $28.

Conclusion

I grant that counting pairs and ace-king combos is relatively simple. But suppose in the heat of battle, a flop comes down and you believe that your opponent could have (among other possible hands) any of the heart flush draws that are two suited Broadway cards, plus all of the ace-high flush draws. How many flush draw combos does she have? (See the answer below.)

Not surprisingly, the best way to get better at this is to practice in the lab (a.k.a. 'your kitchen table'). Go over common situations and learn the arithmetic. Eventually, you'll be as comfortable with the important ones as you're sure that jacks have 37% equity against a range of {QQ+, AK}.

This stuff is not trivial and if you're not used to working with numbers, it can be a bit daunting. But at least some of your opponents are already doing it, and once you get the hang of it, you might even enjoy the mental gymnastics.

P.S. Your villain can have the suited-in-heart combos of A-Q, A-J, A-T, A-9, A-7, A-6, A-5, A-4, A-3, A-2, Q-J, Q-T, J-T, for a total of 13.

Lee Jones can help you count combos and then count your winnings. Go to leejones.com/coaching and schedule a free coaching consultation. Lee specializes in coaching live cash game players.

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  cash game strategytournament strategyno-limit hold'emcombinationscombospot oddsprobabilitiesequityrangesrange readinglive pokeronline poker

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.

How Many Possible Hands In Texas Holdem Poker

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.

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.

Top Hands In Texas Holdem

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.

How Many Cards In Hand Texas Holdem

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%.

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.

How Many Possible Hands In Texas Hold'em

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.

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.