Once upon a time, there was a false dichotomy in the poker world. Some practitioners of the game attempted to divide it into “feel” players and “math” players. It is true that, back in the golden age, some well-known pros relied more heavily on psychology, whereas others took an analytic approach to the game. But closer inspection revealed that the so-called “feel” players could tell you in a heartbeat the odds of hitting a flush draw with two to come. Similarly, the math players simply regarded the psychological elements of poker as another mathematical input parameter.

Do you need to use math to be a winning poker player? Again, the answer is somewhat nuanced. The application of game theory to poker has highlighted that its foundations are deeply mathematical, and that to be on the cutting edge of research in poker theory, you will need to be handy with sophisticated computation. However, if all you want to do is win at poker, you do not need to employ advanced mathematics; you simply need to be able to do a little arithmetic.

In this article, we will outline where this arithmetic plays a critical role in decision making, and provide some short-cuts so that even a “math phobe” can compete on a level playing field with those comfortable with advanced calculation. If you fall into the former category, you will be pleased to know that user-friendly software now makes it possible for you to become a fairly advanced “math player” in a simple, stress-free manner.

If you are very new to poker, we recommend you first check out our Beginner’s Guide To No-Limit Hold’em before jumping in to the remainder of this article.

## Poker Probabilities

Before getting into the nitty gritty of the poker math you need for success, let us first address a more basic question. Why do we need any mathematics at all to succeed at poker?

Perhaps the easiest way to appreciate the necessity of some simple calculation is to recognize that poker, at its theoretical heart, is a wagering game. Whenever we put money into the pot, we are effectively betting that we can recoup that amount of money and more. That is the only way we can profit in the long term.

And “long term” is critical in this context. Clearly when we enter a pot preflop, we are not guaranteed to win the pot and make money, even if we hold AA. But the only justification for committing chips in the first place is the expectation that the situation is profitable for us.

To solidify this idea, consider tossing a fair coin. You will be aware that the probability of it coming up either heads or tails is exactly 50%. The coin is the most rudimentary random number generator, but is good enough to decide who gets the ball first in the Super Bowl, or who bats first in a cricket match.

Suppose someone offered you the following betting proposition on a coinflip. Whenever the coin comes up heads, they give you \$1. When it comes up tails, you give them \$1.20. Crappy deal, huh? In the long term, by which we mean a large number of coin tosses, you are guaranteed to lose money. The probability of the coin coming up heads for you to win is insufficient to justify the odds implicit in the wager.

And this is the connection to poker. Outcomes in poker are probabilistic. If you invest money in a pot without a sufficient probability of winning that pot, you will be a long-term loser at the game. Avoiding that unfortunate outcome means you need to know something about poker probabilities. In the remainder of this article, we highlight what some of those probabilities are, and how you can quickly calculate them at the poker table.

## Poker odds and equity

One of the most fundamental and frequent decisions you have to make in poker is whether you are getting the right “price” to call a bet. You simply have to learn how to get these decisions right if you hope to be a winning poker player. In what follows, we start with some basic principles that more seasoned players may prefer to skip. But if pot odds, outs and equity are currently a closed book to you, this material is crucial.

### Are You Confused By Pot Odds?

In Red Chip Poker’s CORE course, we present all the poker fundamentals needed to play the game. We make it convenient for our subscribers to ask questions on the material via a comments section after each lesson, as well as ongoing discussions on our Discord server. One benefit of this interactivity from our side of the discussion is that it allows us to discover what subscribers find confusing. We have learned the following:

Many people get extremely confused and frustrated by pot odds!

A typical query goes something like this:

I kinda get pot odds, but some people add in the price of the call while others ignore it. Which is correct?

At the risk of sounding like a high school science teacher, the problem is invariably that apples are being compared to oranges. Let us start with a non-poker example to hopefully clarify this persistent confusion.

In the coin-tossing example above, you likely realized the proposed wager was a bad one for us because you know the probability of the coin coming up heads is 50%. Consequently, a fair (albeit uninteresting) bet on a coin toss should pay out the same amount for heads or tails. We can relate this to poker by assuming your opponent bets \$1 that the coin will come up heads, and you call that \$1 bet with \$1 of your own, winning when the toss is tails.

The odds you are being offered on this wager are simply \$1-to-\$1 or 1:1. These are the fair odds corresponding to the “break-even point.” In other words, if you made this bet for a large number of coin tosses, both you and your opponent would break even.

We can also ask the question “what is the probability the coin comes up heads?” If you have had any formal training in statistics, you will be aware that mathematicians express all probabilities as a number between 0 and 1. Thus the 50% chance the coin comes up heads is usually written as a probability of 0.5.

So how do odds of 1:1 and a probability of 0.5 relate to one another? The link derives from the fact that this is a wager. You bet \$1 to win a total “pot” of \$2. \$1/\$2 = 0.5.

The likely reason this simple relation can get obscured in a poker context is that we always have a pre-existing pot to include in such calculations. How do we apply the above ideas to the game of poker?

### The Break-Even Point

Suppose that prior action in a poker hand has put a total of \$100 in the pot. We are heads up against a single opponent who bets their remaining \$50 into us. We have them covered.

Poker math allows us to make sound poker decisions. What is the decision facing us here? Clearly we have the option to call or fold. How do we decide?

First, let us look at the situation in terms of the odds we are being offered. We are required to call \$50 into the \$150 that is in the middle (the \$100 pre-existing pot, plus our opponent’s \$50 all-in bet). Our pot odds are thus \$150:\$50 or 3:1. As in the coin-flipping example, we can ask what contribution we make to the total pot including our call. The answer is \$50/\$200 or 0.25.

This tells us that if we figure to have the best hand here more than 25% of the time, we can go ahead an call. This dividing line between a profitable and unprofitable call is often referred to as the “break-even point.”

It may not be intuitive for you to think about this spot in terms of how often we win it. All the money has gone in, so we either have the winning hand or we don’t! But there are two things to keep in mind. First, we are not clairvoyant. Since we don’t know our opponent’s precise hand, the best we can do is put them on a range, or collections of hands. Some of those hands will beat us, others will not. In this specific example, if less than 75% of our opponent’s hands beat us, we have a profitable call.

The second issue is that many poker decisions are more complex than the simple all-in decision presented above. We may, for example, be drawing to a hand we think will win if we make it. Further, in some scenarios, we may have betting streets left to play that have a bearing on our decision. We will get back to these complexities later.

Our collective experience at Red Chip in coaching players is that some prefer to work in terms of conventional “ratio” odds, whereas others see this as an unnecessary intermediate step and prefer to calculate directly the amount a call is contributing to the total pot. If you “see” odds clearly, but struggle with mental arithmetic, it may be worth memorizing the following common situations.

THEY BET FOR YOUR POT ODDS YOU WANT AT LEAST
Full-Pot 2:1 33% equity
2/3rd-Pot 2.5:1 29% equity
1/2-Pot 3:1 25% equity
1/3rd-Pot 4:1 20% equity

If you’re comfortable with arithmetic, you’ve likely spotted (or derived) the relationship between conventional odds and the fractional form. Specifically, if our pot odds are N:1, then our contribution when we call is 1/(N+1). It’s really that little “+1” in the denominator that is the source of the confusion.

Let us wrap up this section by relating these results to the question posed in the article title. One of the most common poker decisions is whether or not to call a bet. The above discussion demonstrates that calling a bet is not simply a question of determining whether or not we have the best hand. What counts is our probability of winning the hand and its relationship to the price we are being offered.

A critical, and to some player a counter-intuitive consequence of this, is that we will frequently face calling decisions where we figure we will usually lose the hand, but the bet we are facing is small enough that we are compelled to call anyway. Not doing so would cost us money in the long term.

It also follows that good poker players have developed the skill to accurately estimate the probability their hand is good. This ability is based on the skill of hand reading. While good hand readers incorporate multiple pieces of information, mostly focused on the prior action in the hand and information about their opponent’s tendencies, the foundations are again mathematical. We will return to this point in the section “Counting Combos In Poker” below.

We have already hit you with a lot of text, and there is more to come. For those of you who absorb information more effectively via video, here are some of they key concepts explained by SplitSuit:

## Poker Outs And The 4/2 Rule

We have noted that poker math is required to reach objectively-correct, profitable decisions. Any calculation we perform in real time at the table also needs to be simple. The so-called “4/2 rule” is an excellent example of a bit of arithmetic that is both incredibly useful and simple to execute.

You may have heard poker players talk about their “outs” when they’re holding a draw. A flush draw provides a simple example. Suppose we hold 9♥8♥ on a flop of A♥T♥2♦. There are nine remaining hearts in the deck that will make our flush, hence we say we have nine outs. The 4/2 rule tells us the probability we will complete our flush.

Specifically, to find the probability of making our flush on the next card, we multiply our number of outs by two. Since 9×2=18, we have an 18% probability to make our flush on the turn. If we wish to know the probability of making our flush by the river, we multiply by 4. Thus 9×4=36 tells us we have a roughly 36% probability of making our flush when all the board cards are dealt out.

Why is this useful? The utility is easiest to see when we are facing an all-in bet. (We recognize this is far from the only possibility and develop the idea further below.) Let us carry out a similar calculation as before when holding 9♥8♥ on a flop of A♥T♥2♦. There is \$100 in the pot and our opponent who we cover bets their remaining \$100. Our call contributes \$100 to a total pot of \$300, thus we contribute \$100/\$300 ~ 0.33. Therefore, we need 33% equity to call this bet and be profitable. The 4/2 rule tells us we have a 36% probability of making our flush.

So do we call the bet? 36% is above the break-even point of 33%, so at first sight this is a profitable call. But there is a wrinkle. A flush is a strong hand in hold’em, and usually if we make our flush in this scenario, we will win the pot. But what if we can make our flush and our opponent makes an even stronger hand such as a full house?

The most likely way this can happen in the current example is if our opponent has flopped a set. Let us suppose they are holding black aces on that A♥T♥2♦. What is our probability of winning the hand? There are analytical ways of calculating this probability, but most players use a tool like Equilab or Flopzilla that calculates the probability numerically. It turns out our winning probability when facing a set of aces here is about 27%. In other words, if we knew our opponent had top set and they bet all-in for 1/2-pot, we are not getting the right price to call with our flush draw.

In practice, the situation is more complex still. Since we are not clairvoyant, we have no way of knowing if our opponent has flopped a set, or if they hold a weaker hand such as top pair. In such instances, we need to estimate our winning chances against the possible range of hands our opponent may hold. And for the most part, this is not a calculation we will do at the table at all. We will have more to say about off-table work in the penultimate section of this article.

## Implied Odds

It is quite common in hold’em for a pot to be contested by a made hand such as top pair and a draw, as we considered in the previous section. What happens when the drawing hand faces a bet when there is money left to play, rather than the all-in case we have focused on to date?

To put some numbers consider the case when our opponent bets \$50 into a \$100 pot, but also assume they have a further \$100 behind. How do we decide on the correct response to this bet?

We might use the following logic: pot odds require we have at least 25% equity to call this bet. But our odds of making our flush on the turn card are, by the 4/2 rule, only 18%. Should we therefore fold?

Folding in this spot is typically a mistake because it fails to account for our implied odds. The concept of implied odds is a little fuzzier than the concrete notion of pot odds, but stems from the idea that we can make additional money when stacks are not yet all-in.

For example, suppose we call the \$50 bet. The turn fails to fill our flush, and when our opponent shoves their remaining \$100 we do not have the correct price to call, and fold our hand. In this branch of the game tree, we have lost money and it may appear that our flop call was a mistake.

However, the second branch of the game tree sees us making our flush on the turn. If our opponent again bets their remaining \$100 into us, we call and will invariably win that additional \$100. Taken in aggregate, these two possibilities justify the flop call on the basis of implied odds.

Again, you can doubtless see the situation in practice may be more complex still. Will our opponent really bet into us when the obvious flush fills? Rather than the full \$100 left in their stack, maybe we only tease half of that out of them. Or perhaps the idea of calling or folding on the flop was an over-simplification. Maybe a better line was to raise as a semi-bluff?

Even in apparently common scenarios, our decisions can rapidly become quite complex. But if you are not a fan of poker math, this may actually work to your advantage. In practice, these scenarios are analyzed away from the table, and with user-friendly software that involves you merely clicking buttons rather than carrying out the calculations yourself. We will return to this point in the penultimate section.

## Stack-To-Pot Ratio (SPR): A Powerful Flop Metric

Hopefully you are getting the idea that poker math is not some insurmountable obstacle that complicates the game of poker. In fact, when applied correctly, some simple poker calculations can lead us quickly and simply to objectively-sound decisions. An excellent example of such a simple metric is the stack-to-pot ratio, usually abbreviated to SPR.

The first consistent use of SPR seems to have been made in the book “Professional No-Limit Hold’Em,” authored by Red Chip co-founder Ed Miller along with Mehta and Flynn. The definition is perhaps self-evident: SPR is the ratio of the effective (smallest) stack to the size of the pot.

For example, if there is \$50 in the pot on the flop, and the players contesting it have \$150, \$200 and \$300 behind, the SPR is \$150/\$50 = 3.

One can, of course, calculate this ratio on any street. However, on the flop specifically, the SPR provides a remarkably efficient short-cut for us to decide whether or not our hand is strong enough to commit the remainder of our chips. Generally speaking, the lower the SPR, the weaker the hand with which we are prepared to get all-in.

To understand why this is so, it is helpful to relate SPR to the concept of equity introduced above. Suppose the SPR is 1, as would arise with a pot of \$100 on the flop and \$100 behind. Further assume we have a single opponent. We have seen that if we commit our remaining stack, we will contribute \$100 to a total resulting pot of \$300. This would arise if we called a \$100 all-in from our opponent, or if we bet all-in and were called.

Our required equity in such a spot is \$100/\$300 or roughly 33%; that is, if we figure to win this pot at least one third of the time, we have a profitable all-in.

How does this calculation change if the pot was \$100, but effective remaining stacks are \$300? First note that the SPR has climbed from 1 in the previous example to 3. The equity we require to commit in this situations is \$300/(\$300+\$300+\$100) which is 3/7 or roughly 43%.

Had the flop SPR been 0.5, as would apply with \$100 in the pot and \$50 left to play, you might like to demonstrate that our required equity would be a mere 25% to commit.

What you can see from these illustrative cases is that we require a higher equity to commit as the SPR increases. And a higher equity in practical terms means a stronger hand. Thus with an SPR of 3 or below, we can often commit with an overpair or top-pair, top-kicker. At much lower SPRs we are compelled to get all-in with hands as weak as second pair or below. As the SPR increases above 6 or so, we would need a hand significantly stronger than an over pair to commit.

Not only does the SPR provide guidance on how to play our hand on the flop, it also informs our preflop decisions. For example, when we hold two broadway cards, the most likely made hand we can flop is top pair. Consequently, these hands prefer a low SPR.

The flipside is that with speculative hands such as small pairs and suited connectors, we are looking to make sets, straights and flushes. Those hands play better when the flop SPR is high. There are, of course, other considerations that influence our preflop play, but thinking about the kind of SPR your preflop action is likely to create can be helpful in determining our preflop action and the subsequent postflop plan.

## Mental arithmetic tips and tricks for poker

In the comments on our CORE lessons and on our Discord server, we often come across newer players who struggle with the mental arithmetic required when playing poker. Here is a test you might like to try on yourself. If someone asks you “what is eight times seven” do you immediately respond “fifty-six,” or do you start counting on your fingers?

If you come from an educational background in which the times table was drilled into you in grade school, real-time poker arithmetic will be simple for you. If you did not go through that experience, you have a little catching up to do. And part of the problem with giving advice on doing mental arithmetic is that there are so many approaches to carrying out such calculations. After all, it’s “mental” arithmetic, and we can’t see into your mind.

In our numerical examples above, we chose pot and bet sizes that made the subsequent calculation simple. You might reasonably object that real poker situations involve much less friendly numbers. Indeed, for live players, simply tracking the current size of the pot can be a challenge.

If you do play live and experience this difficulty, you will find it gets easier over time. Probably the most important way of accelerating that process is to keep track of the pot size in every hand dealt, even if you fold preflop.

First note that you don’t have to be accurate to the nearest dollar. Here is a common short-cut. You’re playing \$1/\$2. A player opens the pot to \$10 and gets two callers. In detail, the size of the pot on the flop will depend on whether the calls came out of the blinds, and how much rake was swept from the table. But clearly with three players each contributing \$10, the pot is roughly \$30.

You can use this process more generally. Suppose a player opened to \$10, got one caller, then a third player raised the pot to \$40. One of the original players calls the \$40 and the other folds. The pot is dominated by the 3-bet to \$40 and a single call. You know instantly the pot is therefore at least \$80 and can likely add in the extra \$10. The real issue is not that the arithmetic is in anyway complicated, it is that you’re in a noisy casino with flashing lights, trying to make solid poker decisions without spilling your drink all over the table.

So take a deep breath, follow the action, and add up the numbers. Note too that if there has been a lot of action and you’ve simply lost count of the pot size, you can always ask the dealer to “spread the pot.” Again, eyeballing a bunch of red and white chips and concluding the pot is about \$200 is a learnable skill. You can ever practice at home. You don’t have to be Rain Man to do this stuff.

The situation gets a bit trickier when we need to divide large and awkward numbers, but here too a combination of practice and sensible approximations will quickly have you calculating relevant quantities in a heartbeat. One technique is to “bracket” the answer we are after. Here’s a poker example similar to the simplified ones from earlier.

We’ve diligently tracked the pot on previous streets and know there’s around \$185 out there. We face an all-in bet of \$78 and want to know our required equity.

If you are really terrible at mental arithmetic, you may want to round those numbers immediately, but we trust most of you can calculate the total pot in your head: \$185+\$78+\$78=\$341. (I did this by noting \$75+\$75=\$150, adding that to \$185 gives \$335, then adding back the \$6 I knocked off the \$78s brings us to \$341.)

Our required equity is then \$78/\$341. Precisely the kind of awkward numbers that crop up in poker all the time.

A crude way of solving this is to round the numbers to bracket the solution. For example, we know the answer we want is bigger than 70/350, and since 7×5=35 we can see that reduces to 1/5 or 20% equity. Similarly, our required answer is less than \$80/\$320, and given 8×4=32 we can see this is 0.25 or 25%. If you stick 78/341 on a calculator, you’ll find it is about 0.23, roughly in the middle of our bracketed values.

It really does just boil down to practice. And you will find that those old-school, “feel” players who claim not to be “math” players at all can spit out approximations of this accuracy without breaking a sweat.

## Counting Combos In Poker

In this and the following section, we introduce a couple of areas of poker math that represent the transition from in-game calculation to off-table work.

Rather than begin with a formal definition of what a poker combo is, let us kick off with a simple exercise. Suppose an opponent opens preflop, you 3-bet, and the opener 4-bets you. You somehow know that they will only make this play if they hold AK, QQ, KK or AA. Suppose you hold QQ. How many ways can your opponent hold these four different hands?

The number of ways they can hold QQ is the easy one. You have two queens, so there are only two left in the deck. Thus we say your opponent can only have one combo of QQ.

What about the others? For any pocket pair when no cards have been removed by us holding them, there are six possible combos. You might like to check this explicitly with a deck of cards. Thus your opponent can have six combos of AA and six of KK. There are 16 total ways to make AK. Specifically, there are four combos of AKs and 12 of AKo. Again, you can verify this with a deck of cards.

Now suppose that rather than holding QQ you have one of the combos of AK. It doesn’t matter which one. How many combos of the four hand types can your opponent now hold?

Since you don’t have any queens, your opponent now has six combos of QQ. For the other three hand types, however, the fact you hold AK blocks the number of combos your opponent can hold. Now your opponent can only have three combos of AA and KK each, and nine combos of AK.

Information like this can be extremely useful when estimating your equity against an opponent’s range. A fuller discussion is given in this article by SplitSuit.

There is a related reason why the number of combos in a range is important to refine our hand reading; that is, our ability to figure out the range of hands our opponent can hold given the betting action. This is particularly useful for online players, although the diligent live player can also exploit this technique.

Online tracking software gives you a plethora of data for your opponents. One key stat is the preflop “RFI” (raise first in) percentage. It tells you how frequently a player enters the pot with a raise when the action has been folded to them.

Let’s make this more concrete. You find that a specific opponent has a 10% RFI from early position. So what? First, that tells you they play fairly tight, thus they may be competent. But more importantly, that limits the possible hands that they can be playing at all. This is incredibly important as the hand progresses and you attempt to determine your response to their action. Indeed this is the very essence of range reading upon which many poker decisions are based.

So what does a 10% RFI range look like in practice? What combos does it contain?

It is important to note that this percentage form of a range does not reveal the precise range being played. A range has an exact percentage associated with it, but multiple ranges can produce the same percentage form. However, a little card sense can clarify the situation. A player opening a pot with 10% of possible hands is invariably favoring strong hands.

This is another area where working with tools like Equilab and Flopzilla is important. For example, you can use such a tool to determine that the following range correspond to roughly 10% of all possible combos:

77+, ATs+, KTs+, QTs+, JTs, T9s, 98s, AQo+, KQo

Presenting ranges in this way is sometimes referred to as the algebraic form. If you cut-and-paste the above into a range analysis tool, it will display it graphically.

Having assigned an opponent a preflop range based on their stats, the art of hand reading now relies on trimming back that range as subsequent betting action denies the likelihood of some of the original combos. It is not an easy process, but once that can be learned through off-table work.

## Fold Equity: If they fold, you win

Have you heard the claim that aggressive poker is winning poker? Maybe a TV host has pointed out that betting and raising gives us two ways to win the hand. Either we reach showdown with the best of it, or our aggression causes our opponent to fold.

This idea is the basis of fold equity. In detail, the definition of fold equity is a little fuzzy, but the salient point is simply that forcing our opponent to fold means we win money. In fact, you can likely see that an opponent who folds too frequently to our aggression is giving us their chips for free. Such over-folding can produce what is known as an auto-profit spot, in which we can profitably bet or raise any two cards.

In practice, we may need to have some hand equity to make this play profitable. A classic example is the semi-bluff. Suppose we flop a flush draw and our opponent bets into us. We have discussed above how to estimate if we are getting the right price to continue through calling using the 4/2 rule, but in addition to calling we have the option of raising.

How might this be superior? Winning the pot immediately when we only have a draw can be a massively profitable play. The trick is knowing when to do it and how much to bet. And that is where tools like the Red Chip Fold Equity Calculator come in. This allows you to analyze a range of scenarios where forcing your opponent to fold through aggression may be profitable.

There is now a plethora of these web-based calculators that allow anyone — including the math phobes — to probe a range of poker scenarios. Doing so will greatly improve your game. This work essentially primes you for spotting profitable opportunities to earn pots rather than relying on the run of the cards to win you one. A full suite of free calculators is available here.

## Expectation Value

You may have noticed that, in many of the example calculations involving equity, we have restricted our attention to situations in which the last bet is all-in. There is a simple reason for this: when there is money left to play, our current equity need not be a direct measure of our chances of winning the pot.

Consider the classic pair-versus-pair preflop match-up. If we hold AA and our opponent holds KK, the little graphic used by TV shows will indicate we have 80% equity. That is our “fair share” of the pot. And if we could get all the chips in preflop, we would win that pot 80% of the time.

Let’s look at another preflop situation with 100bb starting stacks. Suppose we open 2.5bb from early position with AK and only the button calls. There is now 6.5bb in the middle with 97.5bb left to play.

Exactly how much equity we have in this spot depends on the calling range of our opponent. The only two hands that have us in bad shape — AA and KK — would be expected to 3-bet preflop, so it would appear that equity-wise we are doing well here. In fact if we give the button a typical calling range, we can use a tool like Equilab to calculate that we have around 57% equity. Had we got all-in against that range preflop, we would thus expect to win the pot 57% of the time.

But there is money left play in our current example. Moreover, with multiple betting rounds remaining, we are not particularly interested in how often we win the pot. The important metric is how much money do we win on average? The metric used to answer such questions is expectation value or EV.

You can likely see that this is a far more complicated question. There are many decision points remaining in this hand. If we bet the flop, our opponent can fold, call or raise. If we check, they may check back or bet. We could then fold, call or raise ourselves. And all this before we even consider how much we bet or raise.

Returning to our example hand in which we hold AK, our 57% equity sounds promising, but how much of that equity do we actually realize? A mathematical answer to that question is beyond the scope of this article, and is increasingly being answered using GTO solvers anyway. However, the positional advantage in poker is sufficiently strong that we can anticipate our opponent grabbing more than their “fair share” of pots and chips than would be indicated by raw equity alone.

For more on this complex but important topic, here’s SplitSuit: