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1. Texas Holdem Math Trainer
2. Texas Holdem Probabilities
3. Texas Hold'em Mathematics

• POKER MATH IS EASY TO LEARN. Poker math is a vitally important aspect to No Limit Hold'em, but it is often overlooked or simply not used because many poker players fear it is too difficult to learn. I'm here to tell you it is not. In fact, fundamental poker math is very easy to learn.
• Poker Math & Probabilities (Texas Hold'em) The following tables provide various probabilities and odds for many of the common events in a game of Texas hold 'em. Odds% Example Win% to-1% JJ v 77 80% v 20% A pair will flop four of a kind Two suited cards will flop a flush.

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.

Math is an essential part of the Texas Hold ‘em strategy, and you will not be able to get away from learning the basics. This is where Alton Hardin does a great job. 7 out of 52 means, that although you build your hand using 5 cards, you still have 7 cards from which to choose these 5. In the case of Texas Hold'em, there's the 2 pocket cards and 5 on the board.

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.

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%.

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.

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.

Poker math is such a fundamental aspect of the game that without a basic understanding you’ll be leaking chips every time you play. Fortunately, you do not need to be a math-wiz to grasp the basics. In fact most of the principles require only basic arithmetic, and even the advanced concepts can be understood easily with the help of a poker calculator.

This article introduces the key poker math concepts, starting with the different types of odds. After this you will see the related concepts of outs and equity, which are especially important when it comes to playing draws to flushes and straights. Poker math can be used to help read your opponent’s hand, and I cover 2 ways you can do this in this article. Finally, I have introduced the more advanced concepts including NASH equilibrium and the Independent chip model.

Introducing Odds

If you peel away the layers of complexity involved in poker, you will find the concept of odds at the very heart of this game. Every time you act in poker, you are either taking odds or offering them to your opponents. If you bet $50 into a $100 pot, your opponent needs to call $50 to win $150 more, getting odds or 3-to-1. Instead of assessing his absolute chances of winning the pot, he should assess this in terms of the odds being offered. If he wins more than once for every 3 times he loses (or more than 25% of the time) then this is a profitable call. If he wins less than this, then calling will lose money over time.

Of course, assessing your winning chances is a skill to be learned in its self. What you need to be aware of is that you can’t separate poker thinking from the odds that you take on your bets or offer to opponents.

With multiple streets of betting, the immediate odds (known as ‘pot odds’) are not the only factor. There are many situations where you can win a large amount of chips on future betting rounds. The classic example is when you face a raise holding a small pair before the flop. You are probably behind at that point, though if you catch a set (3-of-a-kind) on the flop, you stand to win a lot of chips.

Calling this raise can be said to have ‘implied odds’, you will hit the set approximately 1 in 9 times (odds of 8-to-1), and so you need to ensure that your chip stack is deep enough to make more 8 times the bet you call before the flop.

The Related Concept of Outs

‘Outs’ are the cards in the deck which will improve your hand. For example, you have 2 spades, and there are two spades and one heart on the flop. You have not seen anyone else’s cards, so there are 47 unseen cards to account for. Since 4 of the 13 spades are out, there are 9 left out of 47. This gives you odds of 47-to-9 (approximately 5-to-1) of hitting a spade on the turn and if you miss this then a similar number (46-to-9) on the river.

As you gain experience at tables, counting outs will become automatic. The real skill comes in comparing the immediate odds and implied odds with the number of outs you have. Bear in mind that when the 3rd spade hits, many opponents will be wary that you have a flush. If you instead hit a straight this is far better hidden – your implied odds will often be higher.

Reading Your Opponent’s Hand

The best poker players are adept at reading their opponent’s hands. This involves estimating their starting hand range, then refining this based on the betting over several streets. It is hard to put someone on an exact hand. However, by the river many people can narrow down the possibilities significantly and even assign weightings based on the likelihood of different combinations.

Math can help with the initial ranges of hands. Some extreme examples will illustrate this point. If a player is super-tight, and only ever raises with 4% of hands, then you can assign the following range:

AA through JJ, Ace-King, Ace-Queen Suited

If they are super-loose and playing 40% of hands, then: All pairs, all aces, most kings and suited queens and many other suited and connected cards are in their pre-flop range.

Texas Holdem Math Trainer

Ideally, you will work this out using a poker calculator when away from the tables. Once you have practiced assigning ranges, you can use that same calculator to show how much equity your hand has against the possible range held by your opponent. With this information, you can then use the pot-odds to decide whether you can profitably play the hand.

Card Distribution

Card distribution can help you assess the likelihood of certain hands within a range. For example, there are 6 ways you can be dealt a pair of aces based on suits. If you instead look at Ace-King, there are 16 ways of this hand being dealt based on the suits.

Texas Holdem Probabilities

Applying this is a matter of assigning a range of hands – then working out the probability of each hand within that range. For example if you think your opponent would raise with Aces, Kings, Queens or Ace-King only in a certain spot, then you know there are 18 ways he could have a pair, and 16 ways he could have Ace-King. You can then work out your equity against these different hands in proportion to their likelihood, and make your odds decision based on this.

Advanced Concepts in Poker Math

Poker math gets more advanced than this, with concepts like the Independent Chip Model (which handles the changing value of tournament chips), SAGE (a system for heads-up play at low blinds) and NASH Equilibrium, which works on a ‘he knows I would play X, so he plays Y, but if he plays Y, I should adjust again and play Z’ system.

Texas Hold'em Mathematics

Poker is a game that will take even experts a lifetime to master. These math concepts will help you make a lot of money from players who are not using them. While you are winning, I recommend you take the time to study the more advanced concepts to increase your edge still further.