No Limit Holdem 101: Holdem Math Part 5 (Putting It All Together)

This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limit holdem cash games.

For the sixth installment of this series, I am going to tackle one of the most important aspects of poker: math.

When I first wrote this article to encompass everything I wanted to discuss about the math of holdem, it was so lengthy it was almost unreadable. So I’m going to break it into several parts.

The fifth math topic I’ll cover is putting it all together.


We are almost experts on holdem math now. We know how to count our outs, how to convert those outs into percentage or ratios, how to calculate pot odds, and how to think in terms of implied odds.
Okay, so not experts, but we are getting a good grasp of the fundamentals.
Now we need to put it all together and actually use math to help us make a decision.
Example one: A flush draw.
We are playing a cash table with $1/$2 blinds. Before the flop we were in the big blind with the Ace of spades and the six of spades (As6s). The action folds to the small blind who raises to $6. We decide to call and put another $4 in the pot bringing the total to $12. Both the blinds have stacks of about $200.
The flop is the Kc7s4s. You strongly suspect that your opponent has paired his King. He leads out with a pot sized bet of $12.

First we count our outs. There are 9 spades that we don’t know where they are and therefore could come to make our nut flush. So we have 9 outs (unless we want to discount our outs for any reason). We also have an Ace with 3 Aces unknown. That gives us 3 more outs. 12 outs times 2 is 24% of making our hand on the turn. That gives us 76 to 24 odds against making our hand. That’s right around 3.1 to 1.

Right now the pot is $24 with us having to make a call of $12. We are getting 2 to 1 to make the call.

Now we need to factor in implied outs. We are chasing a flush and an overcard. Both of these will be easy to see on the board and our opponent will likely shut down and abandon his Kings once the scare card hits the boards. We really can’t add anything to this with implied odds.

So we have 3.1 to 1 against us making the winning hand and we have 2 to 1 for pot odds. Since the odds against making our hand or less than our pot odds, we should not be making this call.

Example two: hidden straight draw with a low flush draw.
Same situation as above expect for the cards we hold. Now we have 5s6s. The action folds to the small blind who raises to $6. We decide to call and put another $5 in the pot bringing the total to $12. Again both players have around $200 stacks.
The flop is the Kc7s4s. You strongly suspect that your opponent has paired his King. He leads out with a pot sized bet of $12.

First we count our outs. There are 9 spades that we don’t know where they are and therefore could come on the turn or river to make our flush. So we have 9 outs. But do we really want to count these as this would make a very low flush. This depends on your read on your opponent. For the sake of argument, let’s say we just know that he’s playing an unsuited King. No need to discount any of our outs. We need to then add another 6 outs to make our straight draw (we’ve already counted the 3s and 8s as spades.) That gives us 15 total outs. So we have a 30% (15 x 2) chance of making our hand on the turn. So our odds are 70 to 30 against us making our hand. That’s 2 1/3 to 1 against making our hand.

Right now the pot is $24 with us having to make a call of $12. We are getting 2 to 1 to make the call.

So based strictly on the pot odds we should NOT be making this call (but it is fairly close).

But now we need to consider the implied odds and see if this makes for a good call or not. If any spade hits the board, our opponent is likely to shut down the betting. If however a non-spade 3 or 8 comes, our opponent is very likely to fire at this pot and we are now getting the right odds to make this call. We are getting no implied odds if we make the flush but high implied odds if we make the straight and that is more than enough add to the pot odds to make this an easy call.

Now you can see why in the last installment I referred to implied odds as more an art than a science. Implied odds require a high degree of predicting your opponent’s actions and that can be tricky indeed.

But wait a minute; isn’t it the rule of two and FOUR? What happened to the freaking FOUR? Good question.

The FOUR applies to your odds of making your hand on both the turn and the river and should be used in all-in situations after the flop. If the flop bet puts either you are your opponent all-in then that is where you use the FOUR as your factor to calculate your percentage of making your hand on the turn and river combined.

Example three: all-in on the flop.
We are playing a cash table with $1/$2 blinds. Before the flop we were in the big blind with the Ace of spades and the six of spades (As6s). The action folds to the small blind who raises to $6. We decide to call and put another $4 in the pot bringing the total to $12. The small blind has $36 left. You have him covered.
The flop is the Kc7s4s. You strongly suspect that your opponent has paired his King. He leads out by going all-in for $36.

First we count our outs. There are 9 spades that we don’t know where they are and therefore could come to make our nut flush. So we have 9 outs (unless we want to discount our outs for any reason). We also have an Ace with 3 Aces unknown. That gives us 3 more outs. 12 outs times 4 is 48% of making our hand on the turn. That gives us 52 to 48 odds against making our hand. That’s right around 1.1 to 1.

Right now the pot is $48 with us having to make a call of $36. We are getting 1.33 to 1 to make the call.

Our pot odds are greater than our odds of making the hands (1.33 is more than 1.1) so we should make the call.

I know some people use the rule of four for the flop regardless, but that is a mathematical error. The fact of the matter is the rule of four is your chance of making the hand on both the turn and the river. In order to use four you will have to calculate both the known bet on the flop and the unknown bet on the turn.

Now this doesn’t mean that you never use the rule of four on the flop unless there is an all-in involved. What it means is that you need to calculate the betting on the turn while you are on the flop in order to use the rule of four to justify your call on the flop. If you suspect your opponent will move all-in on the flop, then you can use those numbers to calculate your pot odds and compare those to your hand odds.

Yes. These calculations can get very tricky on the logic. Even though this about math it is based on your read on your opponent. If you don’t have a read on your opponent, then you are going to be hampered in some of the more complex aspects of holdem math.

Well, that’s it for the math of holdem (except I will be referring back to the math as we discuss further concepts in the future.)
In the next installment I’ll take a step back and talk about my basic theory behind holdem – Giving the other guy the chance to make a mistake.

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No Limit Holdem 101: Holdem Math Part 4 (Implied Odds)

This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limit holdem cash games.

For the fifth installment of this series, I am going to tackle one of the most important aspects of poker: math.

When I first wrote this article to encompass everything I wanted to discuss about the math of holdem, it was so lengthy it was almost unreadable. So I’m going to break it into several parts.

The fourth math topic I’ll cover is implied odds.

In “Phil Gordon’s Little Blue Book”, Phil defines implied pot odds as “A calculation of pot odds based not on the money that’s currently in the pot but on the total money that you anticipate will be in the pot at the end of hand.”

In other words you are trying to predict the betting that will happen in future rounds should you hit your draw and should you miss your draw.

Implied odds are complicated and imperfect. You don’t really know how your opponent will react in future rounds of betting, but you must do your best to incorporate your predictions into your calculations.

One key guideline to remember here is that the harder your hand is for opponent to read, the higher your implied odds will be. If your opponent doesn’t think he’s beat, then he’s more likely to put more chips into the pot.

So, what hands have high implied odds and what hands have low implied odds? Sets generally have high implied odds. You hold the pair and the third card is on the board. A set is one of the hardest hands to put an opponent on. Conversely trips are not a high implied odds hand. You hold one card and the other two are on the board. Your opponent will be wary of your bets since he can easily see the pair on the board.

Straights using connected cards from your hand that form the middle of a straight are higher in implied odds that straights using connected cards from your hand that form the top end of the straight. An opponent looking at a board of 589 will not as readily put you on a straight as an opponent looking at a board of 789.

Flushes are generally the lowest implied odds of any hand. It is easy for your opponent to see the three suited cards on the board and react accordingly.

A quick note on a concept I’ve seen called reverse implied odds. The concept here is that if you hit your draw but you still don’t have the best hand, that you will put a lot of chips in the pot when you are beat. For instance if you Have a suited King and make your flush but your opponent was playing the suited Ace. You have bad reverse implied odds because you made the second nut flush and you may loose a lot if you don’t put your opponent on the nut flush.

This is really is a convoluted way of saying what will happen if I was wrong in counting my outs. Be very careful when counting your outs. It usually isn’t wise to chase 2nd best draws.

Well, that’s it for implied odds.

In the next installment, we will talk about putting it all together and making a decision based on math.

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No Limit Holdem 101: Holdem Math Part 3 (Pot Odds)

This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limit holdem cash games.

For the fourth installment of this series, I am going to tackle one of the most important aspects of poker: math.

When I first wrote this article to encompass everything I wanted to discuss about the math of holdem, it was so lengthy it was almost unreadable. So I’m going to break it into several parts.

The third math topic I’ll cover is pot odds.

In “The Poker Players Bible”, Lou Krieger defines pot odds as “The ratio of the size of the pot compared to the size of the bet the player must call to continue the hand.”
And it really is that simple to calculate simple pot odds. Let’s look at one quick example to illustrate. You are playing a ring table with blinds of $0.05 and $0.10. You are the big blind. The action folds around to the small blind who raises to $0.30. You call the raise. There is now $0.60 in the pot. After the flop the small blinds bets ½ pot ($0.30). The total pot is now $0.90. You have to make a call of $0.30. So you are being asked to put $0.30 into a pot current containing $0.90. Your pot odds are “the ratio of the size of the pot” ($0.90) “compared to the size of the bet the player must call” ($0.30). That’s 90 to 30 or 9 to 3 or 3 to 1.

That’s pretty simple, right?

Well you just know I’m not going to write that short of an article. I’m going to make it more complicated than that just so you have the pleasure of reading more of my meticulously written words. (For the humor impaired, that was an attempt at humor.)

In reality, it is that simple. All I’m really going to add here is that you need to consider all possible betting when you make your decisions. For instance in a three player pot with you being the second player to act, don’t overlook that fact that there is another player to act and his actions can change the landscape. You can’t count on his actions enough to add his call or raise into your mathematical calculations, but you should be aware that the way that player acts could effect the way the hands plays out as well as the size of the pot.

The other important fact to consider is future rounds of betting. Don’t overlook that if you are making this decision after the flop and you make the call that may well be faced with another decision after the turn should your draw fail to hit.

Okay. That’s enough on pot odds or at least this aspect of pot odds.
In the next installment, we will talk about the basics of the next mathematical concept, implied odds.

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No Limit Holdem 101: Holdem Math Part 2 (Converting Outs to Odds)

This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limit holdem cash games.

For the third installment of this series, I am going to tackle one of the most important aspects of poker: math.

When I first wrote this article to encompass everything I wanted to discuss about the math of holdem, it was so lengthy it was almost unreadable. So I’m going to break it into several parts.

The second math topic I’ll cover is making the numbers mean something by converting your outs into percentages or odds.

Now before all of those who got a “C” in algebra start heading for the door, let me tell you that Holdem Math is actually very easy. Holdem Math is as simple as 5th grade math. If you know your times tables, you can do the math needed for Holdem.
In the last installment we focused on how to count outs. Once we know the outs, we need to convert the outs to a number that gives a means of comparison to other numbers. I personally prefer to use percentages instead of odds, but some poker players (being gamblers) prefer expressing this in terms of odds.

The important thing is that you can produce a number that you are comfortable with in order to compare it to your pot odds (discussed in the next two articles.)

We will use a standard flush draw as our example. Lets say you have the Ace of Clubs and the two of clubs (Ac2c) and the flop is the six of clubs, the seven of clubs, and the King of spades. By counting outs we know that we have nine outs to make the nut flush.

But how do we express this in a way that has some meaning? We will use something called the rule of 2 and 4. This rule provides us with an APPROXIMATION of the percentage chance you have of making your hand. If you are trying to calculate the percentage of making your hand after the flop (with both the turn and river to come), multiply your outs by the number 4. If you are trying to calculate the percentage of making your hand on one street (after the flop to make it on just the turn or after the turn with just the river to come), multiply your outs by the number 2.

So in our nine out example above, we would have a 36% (9 x 4) chance of making our flush on the turn and river combined. We would have an 18% (9 x 2) chance of making the flush on the turn by itself and should the spade not come on the turn, we would again have an 18% (9 x 2) of making the nut flush on the river.

A true mathematician will point out that this is an APPROXIMATION. And that is correct the actual mathematically derived percentage based of the number of desired outcomes and the number of remaining unknown cards is 35% to make the nut flush on the turn and river combined, 19.2% to make the nut flush on the turn by itself, and 19.6%. We are trying to use the approximation as a tool so that we can actually do the math in our head on the fly, so we need to be comfortable with the differences.
Since we are working in a very limited universe (remember there are only 169 starting hand combinations) I highly recommend just finding and printing out an outs and odds chart and getting a general feel for the numbers behind the outs and how they compare to the approximations and whether or not you think you need to adjust the approximations.

True gamblers will want to express this percentage as odds (and there is come justification in this as you will need to compare this number to your pot odds, which we will cover in the next part of this series, to help with our decision making.)

So how do we express this as odds? To me this is more complicated math, but here it is and, as always, I’ll try to keep it simple.
Since we are talking percentages, we are always working with a base of 100% and dividing that 100% up in two parts to compare those parts to each other. If we have a 36% chance of making our hand, that means we have a 64% chance of NOT making our hand (100-36=64).

The way to express this in odds is to compare the ratio of the two numbers. The ratio here is 64 to 36 against you making your flush. You’ll need to reduce that down to something versus 1 to have a usable number to compare to pot odds. You can keep dividing each number by 2 until you get close and then you can estimate to have a usable number. 64 to 36 is 32 to 18 is 16 to 9. If this were 16 to 10 then it would be 1.6 to 1 against. It’s a little more than that against so add .2 to make it 1.8 to 1 against. Note here that the actual mathematical computation is 1.77 to 1. Please be careful when estimating as that can produce a number further off than you expect.

That is how to covert your outs into percentages (and odds). Don’t be intimidated by the math of this. The best thing to do is print out a chart and use that until you are comfortable with doing this math in your head.
In the next installment, we will talk about the basics of the next mathematical concept, pot odds.

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No Limit Holdem 101: Holdem Math Part 1 (Counting Outs)

One of the most critical concepts to master in poker is math. We will start out discussion of the math of holdem poker with the basic bulding block of counting outs.


This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limit holdem cash games.

For the second installment of this series, I am going to tackle one of the most important aspects of poker: math.

When I first wrote this article to encompass everything I wanted to discuss about the math of holdem, it was so lengthy it was almost unreadable. So I’m going to break it into several parts.

The first math topic I’ll cover is the starting point for all holdem math: counting outs.

There are 169 starting card combinations in Holdem (if suits are discounted). It is important to know the general value of these combinations of hands. There are many decent starting hands chart that will give you a guide on which hands you should be playing. I’m not going into that in this installment, but I will go into that in a future installment. For this installment, I’m going to stick with the concept of outs.

You will be making your pre-flop decisions based on expected value of starting hands, reads on your opponents, position, etc., but one of the most critical factors in decision making after the flop is outs.
For a simple definition, an out is a card that can come in the community cards that will improve your hand.

As a simple example, if you hold the Ace of hearts and the two of hearts (Ah2h) and the flop is the seven of hearts, six of hearts, and King of spades (7h6hKs) then you have missed your hand (at least slightly) but you need to know what cards might come on the turn and the river to make you a winner. In this example, any heart will give you the nut flush. Since there are 13 hearts in the deck, 2 hearts in your hand, and 2 hearts in the community cards, there are 9 hearts (13-2-2=9) that you don’t know where they are that will make the nut flush for you.

Pretty simple so far, right? 13-2-2=9. Not hard math at all.

Just make sure the outs you are counting are actually outs. The real idea here isn’t to count outs to make just any hand, but rather to count your outs to make the winning hand. If you hold the King of clubs and the three of clubs (Kc3c) and your opponent holds the Ace of clubs and the 2 of clubs (Ac2c), then you can make your flush and still loose the hand. The point of this is to make sure that you are counting outs to a winning hand.

It is also important to understand the concept of discounting outs. There are a couple of different categories that this falls into.

The first is an out that will make you a hand, but will make your opponent a bigger hand. For example, you hold a pair of Queens and your opponent holds a pair of Aces. The flop is King, Jack, Ten. Now obviously if the turn or river is a Queen you will make a set of Queens but your opponent will make a straight (and the bigger hand.)

The important concept here is that you must always consider your opponent’s hand when counting your outs. Ignoring your opponent’s hand will over inflate your calculations and lead to making poor decisions.

The second is a concept that I have not seen expressed very often, but I think it important none-the-less. This is the concept of discounting outs that your opponents may hold in their hand. I think this concept applies almost exclusively to flush draws. If you are counting odds towards a flush draw, you need to consider the very real possibility that one or more of the players at the table was dealt cards in that suit as well. If you are at a 9 handed table (meaning your have 8 opponents) and have suited cards in your hand, then each of your opponents has somewhat less than a 50% chance of holding a card of the same suit as your cards. Against 8 opponents that means the odds say they will hold between 3 and 4 of the outs you need to make your hand. That means the 9 outs you thought you had have suddenly shrunken to 5 or 6 outs after the flop and is not as nearly attractive as the 9 outs.

There are 4 suits in the deck of cards, but there are 13 different ranks. The math gets much murkier if you are trying to deal with the rank of the cards dealt and is really much too complicated to even consider the mathematical possibilities trying to discount outs based on rank.

Some out there may well not agree with the concept of discounting your flush draw based on probabilities that someone holds a card of your suit, but I think that if you are going to use probabilities, then you had best consider all the probabilities (or at least those you can get a handle on).

That’s it for this installment. Count your outs correctly and you are on your way to making holdem math work for you.

In the next installment I’ll deal with calculating the percentages of making a hand and converting those percentages into odds.

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No Limit Holdem 101: Poker is a Situational Game

This is intended to be a series of articles about playing on-line no limit holdem cash games. There will be times where I venture into live poker and times where I venture into SNGs, MTTs, Satellites, and games other than no limit holdem, but for the most part this will target no limt holdem cash games.

For the first installment, I am going to cover a subject that applies to all poker not just no limit holdem and not just cash games.

The first topic is simply situational poker.


I see questions asked all the time along the nature of “How do you play pocket Jacks pre-flop?” and the answer is, “It depends on the situation.”

While there are some general statements that can be made about pre-flop play versus your hand holdings, the single most important aspect of poker is the situation.

The situation is the sum total of the picture at that moment in time. You must take into consideration all possible elements:
• Is this a cash game, a Sit N Go, a Tournament, or a Satellite?
• What is my position at the table?
• What actions have been taken before me in this hand?
• What actions are the players yet to act likely to take based on my action and the actions already taken?
• What are the relative stack sizes?
• What are my opponents’ motivators right now?
• Where am I versus the bubble?
• Where are my opponents versus the bubble?

And there are many, many more that contribute to the current situation.

The point of all of this is that you should spend as much time analyzing the situation as you do the cards in your hands.

Paying attention to the way your opponents act is the key to being able to take advantage of any situation.

Give this a try:
• Depending on your bankroll, go to a cheap SNG or a play chip SNG. Take a sticky note or a piece of tape and piece of paper and put it over your cards.
• Do not look at your cards for the entire SNG.
• Watch your opponents’ play and how it changes over the course of the SNG.
• Try to find situations to make your opponent fold by betting.

This little exercise is something everyone should do from time-to-time just to sharpen their skills in this aspect of the game.

Improve your abilities to read (and take advantage) of the situation and you’ll find yourself benefiting in the long term.

Installment number two will be up to you. I do have an outline with 23 topics, but I’d like to know what topic you would like to see me write on next. Drop me a message, comment on this blog, tweet me, or send up smoke signals. Just let me know what you’d like to see next.

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Change of Direction

I apologize for getting behind on the updates on my play.

I decided to take this blog in a different direction and concentrate on organizing my thoughts on poker into more of an instructions bent.

Starting next week I'll begin post a series I'll be called "No Limit Holdem 101"

Type rest of the post here
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