Texas Holdem Math

This is a very important lesson and can also be quite intimidating to a lot of people as we are going to discuss Poker Math!

• Texas Holdem Probability Chart
• Texas Holdem Probabilities
• Texas Holdem Math Book
• Texas Hold'em Math

Our Free Texas Holdem Poker game is targeted to players with a variety of skill levels. This Free Poker Application is designed to help players learn Texas Holdem without risking their own money as well as players who already know how to play Texas Holdem and want to test out various strategies. 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.

But there is no need for you to be intimidated, Poker Maths is very simple and we will show you a very simple method in this lesson.

You won’t need to carry a calculator around with you or perform any complex mathematical calculations.

What is Poker Math?

As daunting as it sounds, it is simply a tool that we use during the decision making process to calculate the Pot Odds in Poker and the chances of us winning the pot.

Remember, Poker is not based on pure luck, it is a game of probabilities, there are a certain number of cards in the deck and a certain probability that outcomes will occur. So we can use this in our decision making process.

Every time we make a decision in Poker it is a mathematical gamble, what we have to make sure is that we only take the gamble when the odds are on in our favour. As long as we do this, in the long term we will always come out on top.

When to Use Poker Maths

Poker Maths is mainly used when we need to hit a card in order to make our hand into a winning hand, and we have to decide whether it is worth carrying on and chasing that card.

To make this decision we consider two elements:

1. How many “Outs” we have (Cards that will make us a winning hand) and how likely it is that an Out will be dealt.
2. What are our “Pot Odds” – How much money will we win in return for us taking the gamble that our Out will be dealt

We then compare the likelihood of us hitting one of our Outs against the Pot Odds we are getting for our bet and see if mathematically it is a good bet.

The best way to understand and explain this is by using a hand walk through, looking at each element individually first, then we’ll bring it all together in order to make a decision on whether we should call the bet.

Consider the following situation where you hold A 8 in the big blind. Before the flop everyone folds round to the small blind who calls the extra 5c, to make the Total pot before the Flop 20c (2 players x 10c). The flop comes down K 9 4 and your opponent bets 10c. Let’s use Poker Math to make the decision on whether to call or not.

Poker Outs

When we are counting the number of “Outs” we have, we are looking at how many cards still remain in the deck that could come on the turn or river which we think will make our hand into the winning hand.

In our example hand you have a flush draw needing only one more Club to make the Nut Flush (highest possible). You also hold an overcard, meaning that if you pair your Ace then you would beat anyone who has already hit a single pair on the flop.

From the looks of that flop we can confidently assume that if you complete your Flush or Pair your Ace then you will hold the leading hand. So how many cards are left in the deck that can turn our hand into the leading hand?

• Flush – There are a total of 13 clubs in the deck, of which we can see 4 clubs already (2 in our hand and 2 on the flop) that means there are a further 9 club cards that we cannot see, so we have 9 Outs here.
• Ace Pair – There are 4 Ace’s in the deck of which we are holding one in our hand, so that leaves a further 3 Aces that we haven’t seen yet, so this creates a further 3 Outs.

So we have 9 outs that will give us a flush and a further 3 outs that will give us Top Pair, so we have a total of 12 outs that we think will give us the winning hand.

So what is the likelihood of one of those 12 outs coming on the Turn or River?

Professor’s Rule of 4 and 2

An easy and quick way to calculate this is by using the Professor’s rule of 4 and 2. This way we can forget about complex calculations and quickly calculate the probability of hitting one of our outs.

The Professor’s Rule of 4 and 2

• After the Flop (2 cards still to come… Turn + River)
  Probability we will hit our Outs = Number of Outs x 4
• After the Turn (1 card to come.. River)
  Probability we will hit our Outs – Number of Outs x 2

So after the flop we have 12 outs which using the Rule of 4 and 2 we can calculate very quickly that the probability of hitting one of our outs is 12 x 4 = 48%. The exact % actually works out to 46.7%, but the rule of 4 and 2 gives us a close enough answer for the purposes we need it for.

If we don’t hit one of our Outs on the Turn then with only the River left to come the probability that we will hit one of our 12 Outs drops to 12 x 2 = 24% (again the exact % works out at 27.3%)

To compare this to the exact percentages lets take a look at our poker outs chart:

As you can see the Rule of 4 and 2 does not give us the exact %, but it is pretty close and a nice quick and easy way to do the math in your head.

Now lets summarise what we have calculated so far:

• We estimate that to win the hand you have 12 Outs
• We have calculated that after the flop with 2 cards still to come there is approximately a 48% chance you will hit one of your outs.

Now we know the Odds of us winning, we need to look at the return we will get for our gamble, or in other words the Pot Odds.

Pot Odds

When we calculate the Pot Odds we are simply looking to see how much money we will win in return for our bet. Again it’s a very simple calculation…

Pot Odds Formula

Pot Odds = Total Pot divided by the Bet I would have to call

What are the pot odds after the flop with our opponent having bet 10c?

• Total Pot = 20c + 10c bet = 30 cents
• Total Bet I would have to make = 10 cents
• Therefore the pot odds are 30 cents divided by 10 cents or 3 to 1.

What does this mean? It means that in order to break even we would need to win once for every 3 times we lose. The amount we would win would be the Total Pot + the bet we make = 30 cents + 10 cents = 40 cents.

Break Even Percentage

Now that we have worked out the Pot Odds we need to convert this into a Break Even Percentage so that we can use it to make our decision. Again it’s another simple calculation that you can do in your head.

Break Even Percentage

Break Even Percentage = 100% divided by (Pot odds added together)

Let me explain a bit further. Pot Odds added together means replace the “to” with a plus sign eg: 3 to 1 becomes 3+1 = 4. So in the example above our pot odds are 3 to 1 so our Break Even Percentage = 100% divided by 4 = 25%

Note – This only works if you express your pot odds against a factor of 1 eg: “3 to 1” or “5 to 1” etc. It will not work if you express the pot odds as any other factor eg: 3 to 2 etc.

So… Should You call?

So lets bring the two elements together in our example hand and see how we can use the new poker math techniques you have learned to arrive at a decision of whether to continue in the hand or whether to fold.

To do this we compare the percentage probability that we are going to hit one of our Outs and win the hand, with the Break Even Percentage.

Should I Call?

• Call if…… Probability of Hitting an Out is greater than Pot Odds Break Even Percentage
• Fold if…… Probability of Hitting an Out is less than Pot Odds Break Even Percentage

Our calculations above were as follows:

• Probability of Hitting an Out = 48%
• Break Even Percentage = 25%

If our Probability of hitting an out is higher than the Break Even percentage then this represents a good bet – the odds are in our favour. Why? Because what we are saying above is that we are going to get the winning hand 48% of the time, yet in order to break even we only need to hit the winning hand 25% of the time, so over the long run making this bet will be profitable because we will win the hand more times that we need to in order to just break even.

Hand Walk Through #2

Lets look at another hand example to see poker mathematics in action again.

Before the Flop:

• Blinds: 5 cents / 10 cents
• Your Position: Big Blind
• Your Hand: K 10
• Before Flop Action: Everyone folds to the dealer who calls and the small blind calls, you check.

Two people have called and per the Starting hand chart you should just check here, so the Total Pot before the flop = 30 cents.

Flop comes down Q J 6 and the Dealer bets 10c, the small blind folds.

Do we call? Lets go through the thought process:

How has the Flop helped my hand?
It hasn’t but we do have some draws as we have an open ended straight draw (any Ace or 9 will give us a straight) We also have an overcard with the King.

How has the Flop helped my opponent?
The Dealer did not raise before the flop so it is unlikely he is holding a really strong hand. He may have limped in with high cards or suited connectors. At this stage our best guess is to assume that he has hit top pair and holds a pair of Queens. It’s possible that he hit 2 pair with Q J or he holds a small pair like 6’s and now has a set, but we come to the conclusion that this is unlikely.

How many Outs do we have?
So we conclude that we are facing top pair, in which case we need to hit our straight or a King to make top pair to hold the winning hand.

• Open Ended Straight Draw = 8 Outs (4 Aces and 4 Nines)
• King Top Pair = 3 Outs (4 Kings less the King in our hand)
• Total Outs = 11 Probability of Winning = 11 x 4 = 44%

What are the Pot Odds?
Total Pot is now 40 cents and we are asked to call 10 cents so our Pot odds are 4 to 1 and our break even % = 100% divided by 5 = 20%.

Decision
So now we have quickly run the numbers it is clear that this is a good bet for us (44% vs 20%), and we make the call – Total Pot now equals 50 cents.

Turn Card

Turn Card = 3 and our opponent makes a bet of 25 cents.

After the Turn Card
This card has not helped us and it is unlikely that it has helped our opponent, so at this point we still estimate that our opponent is still in the lead with top pair.

Outs
We still need to hit one of our 11 Outs and now with only the River card to come our Probability of Winning has reduced and is now = 11 x 2 = 22%

Pot Odds
The Total Pot is now 75 cents and our Pot odds are 75 divided by 25 = 3 to 1. This makes our Break Even percentage = 100% divided by 4 = 25%

Decision
So now we have the situation where our probability of winning is less than the break even percentage and so at this point we would fold, even though it is a close call.

Summary

Well that was a very heavy lesson, but I hope you can see how Poker Maths doesn’t have to be intimidating, and really they are just some simple calculations that you can do in your head. The numbers never lie, and you can use them to make decisions very easy in Poker.

You’ve learnt some important new skills and it’s time to practise them and get back to the tables with the next stage of the Poker Bankroll Challenge.

Poker Bankroll Challenge: Stage 3

• Stakes: $0.02/$0.04
• Buy In: $3 (75 x BB)
• Starting Bankroll: $34
• Target: $9 (3 x Buy In)
• Finishing Bankroll: $43
• Estimated Sessions: 3

Use this exercise to start to consider your Outs and Pot Odds in your decision making process, and add this tool to the other tools you have already put into practice such as the starting hands chart.

Texas Holdem Guide Part 2

Welcome! This is part 2 of our 4 part guide where we take you from not knowing a thing about Texas Holdem to the point to where you can sit down, play your first game and possibly make some money.

Texas Holdem Guide » Part 1 » Part 3 » Part 4

In the last section I covered the basics to Texas Holdem including the rules, what hands beat what, and the different betting formats. I discussed betting in more detail, too. I basically covered all the details needed for you to be able to sit down and play your first poker game.

The plan for this next section is to dive into some basic poker strategy. Concepts that all beginner players should know, especially if they want to stop losing money and become winning players.

The specific topics I’m going to cover include table images and how they affect your strategy. Then I’ll get into some poker math. But don’t worry; it won’t be difficult to understand. But it will add some much needed structure to your game.

Let’s get into it.

Understanding Table Image

Table image is an important concept to understand.

Table image is how you’re perceived by the other players at the table. Your image is determined by how often you play hands and how you choose to play them. So frequency and aggression, or lack thereof. Other factors come into consideration, too, such as history, mood and previous hands.

The reason why table image is so important to understand is because it will affect how you play against as opponent, as well as how your opponents play against you. If you can label an opponent with an image, you can get into their head and know how they’re going to play certain hands. With this knowledge you can think a level above and out play them.

Before I get into any strategy, though, lets go over the common table image “labels.”

Types of Table Images

• Tight – This player opens few hands. Also known as a nit.
• Loose – This type of players opens a lot of hands.
• Aggressive – Aggressive players open-raise and re-raise more often then calling and checking.
• Passive – Passive players prefer to call and check more often than raise and re-raise.

When labeling a player you’ll want to do so using a combination of my list above (using 2 of the 4 table nuances). In other words:

• Tight/Aggressive – This player opens few hands. The hands they do play are high value and raised or re-raised more often than not.
• Tight/Passive – This type of player opens few hands, and will limp, call or check when they do.
• Loose/Aggressive – This opponent will play a lot of hands, and raise or re-raise most of them. This player can come across as reckless.
• Loose/Passive – Plays a lot of hands, but calls, checks or folds often.

Table images aren’t static either. Table images are dynamic. They can change on a dime, either from good players switching gears and adapting to their opponents, or maybe due to multi-tabling, variance/tilt, mood, and so on.

The best way to determine a player’s image is to simply pay attention. How many hands does your opponent play? How do they play their hands? What do they showdown?

However, it’s not always possible to pay close enough attention to know what someone’s image is. In these cases you can use other clues to help you out:

• Check Sharkscope for your opponent’s stats.
• What is your opponent’s VIP status?
• Are your opponents multi-tabling?
• Do you know if your opponent participates (regularly) in forums?

Texas Holdem Probability Chart

As a rule of thumb, if your opponent has (good) stats, high VIP status, multi-tables and participates in forums, they are probably a good player (or trying to become one). The recommended strategy for good players (or to become a good player) is to be tight aggressive. Open high quality hands, and put a lot of money into the pot with them.

This isn’t always the case, but it just gives you something to go off of if you’re not sure about an opponent or two at your table.

How Image Will Affect Your Strategy?

Ok, so now that we know what image is and how to determine the type of player someone might be, lets look at how this might affect our strategy.

The rule of thumb is to do the opposite of your opponent. In other words, if your opponent is a maniac (reckless, loose-aggressive player), they are opening a lot of hands. So your strategy should be to play fewer hands. To be clear, though, that means fewer hands than them, but possibly (ok, more likely) more hands than you usually play.

To give you an example, say your opponent opens more than 50% of their hands. You decide to play K9o and hit top pair on the flop (rainbow). Against your tighter or average player, you probably wouldn’t play K9o, let alone call too many bets with it postflop. However, against a maniac it’s the equivalent to playing KQ or AQ against an average or good player. So you should treat it as such. If your opponent bets, you call and plan on showing down your top pair.

Lets look at another example. Say you open-raise with KQ and an opponent that opens less than 10% of hands re-raises you. Think about it for a second. What type of hands does someone like this play, let alone re-raise? Aces, kings, queens and maybe AK. All better hands than KQ, so it makes sense to fold.

Do you see what I’m getting at here? Your opponent’s table image will give you an idea as to what hands they play. Then you take this information and determine how good your hand is, if it’s likely to beat theirs and then play accordingly.

Keep Your Table Image in Mind Too

It’s important to keep your own image in mind, too. Have you opened a lot of hands lately? Did you raise them all (like you probably should have)? Or have you gone the last 3 orbits without a hand worth playing? You need to be aware of these things so that you know what to expect from your opponents when you do decide to play your hand. You can use your table image to your advantage, too.

For example, say you haven’t played the last two orbits. You’re on the button, everyone folds to you and you look down to see 96s. Not a great hand, but it’s suited and connected. More importantly, though, is that you haven’t played a hand in so long that if you open-raise here, the blinds will think that you actually have a hand. They’ll be more likely to fold. The bonus (if you want to call it that) is that if they do call, your 96s can hit the flop hard (and be disguised), it’s easy to play postflop and you’ll be in position.

On the other hand, if you open-raised the last 7 hands, you might want to reconsider opening the next hand while holding JTo. At some point you have to expect that someone will get tired of you “bullying the table” and take a stand. In some cases you can use this to your advantage, but in others it’s just too close a call to bother with. So you’re better off folding and preserving or improving your table image.

That’s really all there is to table image and strategy. I hope you see the importance of table strategy and understand how to label your opponents and adjust your strategy for each type of player. It’s real simple, yet many players don’t pay attention or care enough to use it. Don’t be this player.

Poker Math for Beginners

What I’m going to cover in this next section is some poker math. Don’t worry, it won’t be as hard as you might think. So long as you can add, subtract, multiply and divide small numbers, you should be fine.

The following math concepts cover poker outs and pot odds. These are important to learn because they will teach you how to profitably draw to hands. You’ll see why drawing to gut shot straight draws are a bad idea, and how to choose your spots for when to draw to flushes and open-ended draws.

So with that in mind lets get right into it. The first thing I’m going to cover are poker outs.

Poker Outs

Lets start with defining poker outs. What is an out?

Poker outs are cards left in the deck that can improve your hand. For example, say you hold JTo and the flop is A-K-2 rainbow. A queen would improve your hand, so that would be an “out.” Since there are 4 queens in a deck of cards, you would have 4 outs.

Lets look at another example. Say you have AK of hearts. The flop is Q-8-3, two hearts and one club. You (should) know that there are 52 cards in a deck, 13 cards of each suit. Since you can see 4 hearts, basic subtraction will tell us that there are 9 cards left in the deck that will improve our hand to a flush, or 9 outs. In this case an ace or king would improve our hand, too, so if we assume there are 3 aces and 3 kings left in the deck, that means we actually have 15 outs to improve our hand.

Simple. Right?

Now before I move forward, there are 3 important points I want to make:

1. You should only discount the cards you can see. In my example above I don’t know if one of my opponents has an ace or king. Since I don’t know for a fact, I just assume they’re all available, as opposed to trying to discount them.
2. Don’t count outs to hands that won’t likely be the best at showdown. In other words, in my first example with JT, I didn’t count the jack or ten as an out because with an ace or king on the flop I don’t think jack or ten high will be good. However, in my AK example I think making a pair of aces or kings can be the best hand, so I count those as outs.
3. Don’t double count your outs. If the ace of spades will make you both a pair and a flush, don’t count that as two outs. That’s still only one out, even though it can make two hands.

Ok, so now we know what outs are, how to count them and what not to do. This is only half the battle. We now need to convert our outs into odds.

Converting Outs to (Card) Odds / Percentages

The next step is to take our outs and turn them into odds. There are two ways to calculate card odds in poker.

The Rule of 2 and 4

The rule of 2 and 4 is the easiest and fastest way to take your outs and turn them into odds. What you do is this —

If you want to know the odds of hitting your hand over one street (flop to turn, or turn to river) multiply your outs by 2. Then add 1. For example, if you have 8 outs, then the odds of hitting your hand on the next street is 17% ( (8*2) + 1). If you want to know your chances of hitting your hand by the river (from the flop), multiply your hand by 4 and add 1. With 8 outs this would be approximately 33%.

Note: Believe it or not, this method is very close, usually within 1-2%. That’s close enough, too, because if you’re worried about that margin of error you’re chasing hands you shouldn’t be.

Texas Holdem Probabilities

Do you see how easy this is? It takes seconds to do. The rule of 2 and 4 is much better than the next method which is…

Doing the Math

The other way you can figure out your card odds is to do the math (the long way). I don’t recommend doing this while playing because it takes too long to do. I’m only sharing it because I think you might find it interesting or useful.

So what you do is this — take the number of cards you see and subtract that from the total number of cards in a deck. So, on the flop you would know 5 cards — your 2 hole cards and the 3 community cards. So we subtract 5 from 52, giving us 47.

Then using a calculator, what we do is take the number of outs we have and divide that by the cards left in the deck. Using my example above, take 8 and divide that by 47. You’ll see that gives us 17%, which is for one street, and matches what I said above. Then multiply that by 2 for two streets, which is 34%, just slightly more than my example above.

As you can see, this way is more time intensive to do.

But that’s all there is to poker outs; how to count them and turn them into odds and percentages. Keep this in mind as we move to the next section because we will need to compare these card odd percentages to our pot odds. That will determine whether or not we’re drawing profitably to a better hand.

Pot Odds

Like the last section, lets start with defining what pot odds are.

Pot odds is the number you come up with when you do the math between what is in the pot and how much you have to invest to win it.

Pot odds are important because you’ll compare them to your card odds to determine whether or not it makes sense to draw to a better hand. You may also use pot odds in tournament or sit and go situations where your opponent shoves all in, you determine their range (a more advanced topic) and you need to compare your pot odds to their hand range to determine if it’s profitable for you to call them.

The bottom line — pot odds are important. Pot odds are easy to calculate, too.

How to Calculate Pot Odds

Texas Holdem Math Book

Calculating pot odds is simple to do. I’ll explain it using an example so you can see how it’s applied to a real situation.

Say you are in a hand, in position, with one other player. There are 250 chips in the pot, the flop is A-K-4 rainbow and you have JTs. Your opponent bets 100 chips. To figure out your pot odds you just look at the ratio of money in the pot to how much you need to call to win it. In this case, the ratio would be written out as 250 to 100. To make your pot odds easier to use, however, you’ll want to find a common denominator and simplify the ratio as much as possible. So, 250 to 100 turns into 2.5 to 1.

However, pot odds are going to be easier to use if they’re in the form of a percentage. Remember that’s how our card odds from the last section are written. To do this, you’ll need to take what you need to invest, and divide that by the total you stand to win (the pot + your investment). Our equation would look like this — 1/3.5. This comes out to 28.5%.

Note: This will be hard to do in-game, so once I show you how to use this information, I’ll show you a few common pot odd percentages to memorize.

Alright, so lets use this information.

Using my example above, lets decide whether or not it makes sense to draw to our gut shot draw. The only card we can be sure will improve our hand is a Q, so we know we have 4 outs. From the flop to the turn (all we’ll worry about for now), we know that we’ll make our hand about 9% of the time.

We’ve already figured out our pot odds — 28.5%.

To (quickly) determine if the hand you’re drawing to is profitable or not, just compare the two percentages. If thepercentage that you’ll make your hand is higher than your pot odds, then it’s profitable. However, if it’s smaller than your pot odds, like in this case, then it’s not.

That’s all there is to it. There is further math you can do to see how this works, like calculating equity over the long run. But I’ll leave that for you to play around with.

Pot Odds worth Memorizing

I mentioned a second ago that it will be easier for you if you memorize common pot odd percentages. Memorize the following odds/percentages and you’ll find life a little easier — at least at the poker tables.

Pot odds of…

• 1:1 – 50%
• 1.5:1 – 40%
• 2:1 – 33%
• 2.5:1 – 28.5%
• 3:1 – 25%
• 4:1 – 20%

That’s all there is to it. So, to summarize the math section of this guide:

• Outs are the cards left in the deck to improve your hand.
• The rule of 2 and 4 is the fastest way to figure out your chances of improving.
• Turn your fractions / card odds into percentages.
• Compare your card odds to your pot odds. If your card odds are higher, you can profitably call, and if they’re smaller, then it’s better to fold.

Texas Hold'em Math

There are exceptions of course, like implied odds, worrying about stack sizes, history, hand reading and so on. Those are advanced topics, though, which are for another day. This is what I’d stick to for now. It’ll keep you from making the most common errors that beginners tend to make (like chasing gut shot draws).

Conclusion — End of Part 2 of Our Texas Holdem Guide

So that wraps up the second part of our guide. Our first section took you from knowing nothing to the basics of how to play Texas Holdem, and this part just took you from a losing player to (at least) a breakeven player.

You’re welcome ;)

In the next section I’m going to cover more important concepts. Things like table position, bluffing tips for beginners and the basics to playing from the blinds. Concepts that tend to trip beginners up and cost them a lot of money.