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Read More# Poker Math: The Numbers Behind Winning Strategies

## Math in Poker

When people see the word “math,” their immediate reaction is to run away scared, thinking that they left that behind when they graduated high school. We’re here to show you that poker math isn’t as scary as it looks, and once you get to grips with it, you’ll see just how powerful it is.

## The Foundation of Poker Math

So, when people say poker math, what exactly do they mean? Well, there are many different aspects of math that can be used in poker that form the foundation of a winning player’s strategy. There’s probability, which allows you to work out the likelihood of an event happening, which is useful for figuring out whether or not you’ll make your draw. Coupled with that you have pot odds, that allow you to work out how often you need to win for a call to be profitable.

You also have expected value (EV), which allows you to work out the profitability of any given action – the best players are constantly running EV calculations to ensure that the plays they’re making are profitable. These are the core mathematical concepts that you should be using in poker, and we’ll be covering each of them in detail in this article.

## Odds and Outs: Calculating Your Chances

One of the first things that you should learn when learning poker math is pot odds and poker outs; or, to put it another way, how often you will make certain hands, and how to work out if you can profitably draw to them.

When you face a bet from your opponent, you’re given a price to make the call. By comparing the amount that you have to call to the size of the pot, you can easily work out how often you need to win to make the call profitable. This is called calculating pot odds; let’s take a look at the formula for calculating it.

**Pot Odds = Pot Size:Bet Size**

Pretty simple, right? Just to make sure you understand, we’re going to plug in some numbers to that formula using a real-world example; our opponent has just bet $10 into a $10 pot on the flop, what are our pot odds?

**Pot Odds = 20:10 = 2:1**

Having simplified the pot odds, we can see that we’re getting 2:1 on a call. But, how does that help us in-game? Well, we can convert that ratio into a percentage that tells us how often we need to win for the call to be profitable. Let’s take a look at that equation.

**Pot Odds Percentage = (Size of call / (Size of pot + Size of call)) * 100**

Hang on, I thought this was meant to be easy math! Don’t worry, the equation looks a lot more complicated than it is. If we apply a real-world scenario to this equation, you’ll quickly see how easy it really is.

We’re on the flop and our opponent makes a bet of $10 into a $10 pot. How often do we need to win for it to be a profitable call? Let’s plug some numbers into our equation and find out.

**Pot Odds Percentage = ($10 / ($20 + $10) * 100**

**Pot Odds Percentage = $10 / ($30) * 100**

**Pot Odds Percentage = 0.33 * 100 – 33%**

So, you can see that we need to win at least 33% of the time to make the call.

But, how do we know if we win 33% of the time? One of the ways we can do this is by calculating the number of outs we have to make our hand. This is most commonly used when holding a drawing hand, such as a straight draw and flush draw, but it can also be used to include pair-outs as well.

To work out the likelihood that we make our hand, we need to work out how many outs we have and how many unknown cards are left in the deck. Let’s say that we have 5h4h on a flop of AhKd9h. We have a flush draw, and we assume that only a heart will give us the best hand.

We can quickly work out that we have nine outs, as there are nine remaining hearts in the deck and there are a total of 47 unknown cards (52 – our 2 hole cards – the 3 flop cards). To work out how often we will make our hand on the next card, we need to divide our outs by the number of cards left in the deck, in this case, it would be 9/47, which equals 0.191 or 19.1% of the time.

So, with this hand against our opponent’s $10 bet, we know that we’re not getting the right odds to call, so we can comfortably make the fold!

But what if that $10 was an all-in bet and we were guaranteed to see the turn and the river, does that change things? Well, we have another opportunity to make our flush on the river, so let’s work out the possibility of that happening and see how it changes the likelihood that we’ll win.

We’ve already worked out that we will make our flush 19% of the time on the turn, but things change slightly when working out the odds of making our flush on the river. If the turn isn’t a heart, we know that one more card has been removed from the deck, and there are still nine cards that make our flush.

This means that to calculate the odds of making our hand on the river, we need to divide our 9 outs by the remaining 46 cards in the deck, rather than the remaining 47 cards. So, when we work out 9/46, we see that we get 0.195 or to put it another way, we’ll make our hand 19.5% of the time.

However, it’s not as easy as adding the two percentages together and going on your merry way – the actual equation is a little more complicated, so let’s take a look at it.

**Odds of hitting on turn or river = % turn hit + (% river hit * (1 – % turn hit))**

Looks scary, right? Well, once you add some numbers to it, it’s not as hard as it looks. Let’s use our example above to see how often we would make our flush on the turn or river.

**Odds of hitting on turn or river = 0.191 + (0.195 * (1 – 0.191)**

**Odds of hitting on turn or river = 0.191 + (0.195 * 0.809)**

**Odds of hitting on turn or river = 0.191 + 0.157**

**Odds of hitting on turn or river = 0.348 = 34.8%**

As you can see, the odds of hitting our flush on the turn or river are 34.8%, so if we’re guaranteed to see both cards, we’d be making a profitable call!

## The Rule of 2 and 4: Quick Pot Odds

If all that seems a little complicated for your liking, there is a handy shortcut that gives you a good approximation of the equations we’ve learned above. It’s called, the 2/4 rule. If you want to calculate your chance of hitting the card you need on the turn or river, you don’t need to do complicated division to get your answer, just follow this simple rule.

If you want to work out how often you will hit one of your outs on the next card, simply multiply your outs by two, and if you want to work out how often you will hit one of your outs on the turn or river, simply multiply your outs by four.

So, to take our above example, if we want to work out the odds of hitting our flush on the turn, all we need to do is multiply our 9 outs by 2, which gives us 18%. Then to work out the odds of hitting our flush on either the turn or the river, we multiply our 9 outs by 4, which gives us 36%

While this won’t give you a precise answer, it’s a close enough approximation that allows you to do the math quickly, allowing you to effectively use it in-game.

## Implied Odds: Factoring in Future Bets

While pot odds are all well and good, all they do is allow us to work out the profitability of calling on a certain street, they do not take into consideration further action where we could win an exponentially bigger pot. Luckily, there’s a tool that we can use to calculate whether a call would be profitable based on future action. Enter, implied odds.

Implied odds are so named as they consider implied betting action on further streets. So, if you have a hand where you’re not getting the right immediate odds to call, but you think you’ll get action on later streets if you hit your hand, you can use implied odds to see whether or not the call will be profitable given that further action.

Many people use implied odds without actually realizing there’s math behind it; they simply guess whether or not they will get more action on later streets, and if they think so, they’ll make the call even if they’re not getting the right odds. This is sub-optimal. Following the equation we’re about to show you, you can calculate exactly how much you need to make on later streets for your call to be profitable.

**Implied Odds = ((1/% Chance you improve) * Amount to call) – (Current pot size + Amount to call)**

They always look scarier with words, don’t they? Let’s get some numbers plugged in and see how that looks in the real world. For our example, we’ll stick with our $10 bet into a $10 flop while we’re holding a flush draw – we’ll even use the equity calculation from the 2/4 rule to simulate what it would be like if you were to use it in-game.

**Implied Odds = ((1/0.18) * $10) – ($20 + $10)**

**Implied Odds = (5.55 * $10) – $30**

**Implied Odds = 55.5 – $30 = $25.50**

Given the amount we have to call and the likelihood that we’ll improve, we can see that we need to make at least $25.50 on later streets to make this call profitable. Having an actual dollar amount that you can use to estimate your implied odds will make your decisions more accurate, as you’ll know your breakeven point and will be better able to gauge how often you’ll make it.

## Expected Value (EV) and Its Applications

All of the equations that we’ve looked at so far have been about whether or not it’s profitable to call. However, if you want to take things one step further, you can work out the actual value of a call, fold, or raise.

When playing poker, it can be easy to get caught up in the results a hand; if you won, you made a good decision, if you lost, you made a bad decision. However, things don’t work like this in poker, and it’s perfectly possible to make money from a bad decision and lose money from a good decision. To determine whether or not the decision was good, we need to look at how much we’d expect to make if we made the same decision thousands of times over. For this, we use a concept called expected value.

As the name suggests, this concept takes four different data points to give you the *expected* value of a decision; those data points are the amount of money you make if you win, the percentage of the time you win, the amount of money you lose if you lose the hand, and the percentage of the time you lose.

Let’s take a look at how those data points are combined into an equation.

**EV = (How often you win * Amount you win) – (How often you lose * How much you lose)**

Even with words, this one looks simple enough, doesn’t it? Let’s use a couple of examples to see it in action.

In our first hand, our opponent shoves for $10 into a $10 pot on a board of AhKc9h, and we hold the 5h4h. We know from our earlier segments that we can profitably call this bet, but what’s our expected value of making this call?

**EV = (0.36 * $30) – (0.64 * $10)**

**EV = $10.80 – $6.40**

**EV = $4.40**

Using this equation, we can see just how profitable our call is!

But what if we want to work out the profitability of a bet? How can we use the equation for that? Well, consider this scenario, we’re on the river with our 5h4h, and we’ve made our flush. The board reads AhKc9h7s2h – we a flush, and there’s $40 in the pot. We want to work out the EV of two different bet sizes – a pot-size bet, and a half-pot bet. To do this, we have to make some assumptions.

While we likely have the best hand, there’s a small chance that our opponent has a higher flush, so let’s say we win 95% of the time with our hand. We also need to make some assumptions about our opponent’s calling rate – against a half-pot bet, we think they’ll call 70% of the time, but against a full-pot bet, we think they’ll only call 40% of the time.

To work out which of the more bet sizes is profitable, we need to use a more complex EV equation.

**EV = (% Fold * Current Pot) + (% Call * ($ Won * Win %)) – (% Call * ($ Lost * Lose %)**

While this one looks scarier, it’s just adding in the percentage of the time our opponents fold and we win what’s already in the pot.

Let’s look at it in action for our two examples, starting with the half-pot bet.

**EV = (0.3 * $40) + (0.7 * ($60 * 0.95)) – (0.7 * ($20 * 0.05))**

**EV = $12 + (0.7 * $57) – (0.7 * $1)**

**EV = $12 + $39.90 – $0.70**

**EV = $51.20**

So, we can see that the expected value of betting half pot is $51.20, let’s compare that to betting full pot.

**EV = (0.6 * $40) + (0.4 * ($80 * 0.95)) – (0.4 * ($40 * 0.05)**

**EV = $24 + (0.4 * 76) – (0.4 * $2)**

**EV = $24 + $30.4 – $0.80**

**EV = $53.60**

As you can see, the EV of betting full pot is marginally more profitable than betting half pot, so even though our opponent will fold more often, we’ll make more money in the long run with a pot-size bet.

## EV and Bluffing: The Rational Bluff

But what about if we’re bluffing? Can we use EV calcs to work out the profitability of a bluff? Yes, we can! We can run calculations similar to these to find out if our bluff will be profitable based on the percentage of the time we think our opponent will fold to our bet. There are two equations we can use, one for when we’re bluffing with zero equity, and one for when we’re semi-bluffing.

If we want to work out the EV of a zero-equity bluff we need to use this equation.

**EV = (% Fold * Current Pot Size) – (% Call * Bet Size)**

In simple terms, we look at how often we win the pot when our opponent folds, and how often we lose our bet when our opponent calls. So, let’s say that in the previous hand, we didn’t make our flush, and we’re left with five-high on the river. We decide that we need to bluff if we want to win the hand, and we still think that our opponent will fold 60% of the time if we make a pot-size bet. What’s the EV of our bluff?

**EV = (0.6 * $40) – (0.4 * $40)**

**EV = $24 – $16**

**EV = $8**

Our EV of bluffing for a pot-size bet is $8, making bluffing a profitable decision.

But, what if we have equity when we’re bluffing? How can we factor that into our EV?

Well, we can reuse the more complex EV calculator to include how often we’ll make the best hand on the river. So, let’s say that we’re on the turn with our 5h4h, we’re betting with our flush draw, and we still think our opponent will fold 60% of the time to a pot-size bet. What’s the EV of a pot-size bet, given that we have a chance to make the best hand by the river?

**EV = (0.6 * $40) + (0.4 * ($80 * 0.18)) – (0.4 * ($40 * 0.82)**

**EV = $24 + (0.4 * $14.40) – (0.4 * $32.80)**

**EV = $24 + $5.76 – $13.12**

**EV = $16.64**

As you can see, just by having 18% equity, the EV of our bluff has more than doubled! This shows the power of semi-bluffing, and why players always try to bluff with equity whenever possible.

## Equity and Equity Realization: The Bigger Picture

We’ve spoken a lot about equity in this guide to poker math, so it’s about time that we explain what it is. Every poker hand has a percentage of the time that it will win the pot; it could be 0%, 100%, or anywhere in between. This percentage is often referred to as its equity, or in other words; its share of the pot. If we split the pot based on the likelihood that each hand would win, the amount that you would receive would be equal to the equity percentage, so if your hand has 20% equity, you’re entitled to 20% of the pot. So, when you fold a hand that has equity, you’re folding part of the pot that’s technically yours.

Now, this doesn’t mean that you should see every hand until the river. What it does mean is that you should take steps to realize as much of your equity as possible. When a hand has reached the river, we know for certain who has the best hand, and that hand will have 100% equity. However, until then, we’re unsure who will win the pot, and it’s often the case that multiple hands have a chance of winning.

This is part of the reason why being in position is so powerful in poker. If everyone checks to you, you get the final say on whether or not the next card is dealt. You can decide to check back and realize your equity, or you can bet and potentially deny your opponents their equity. Sometimes, if your hand has a reasonable amount of equity, you may want to play passively to try and realize that equity rather than bet and potentially get raised off of it.

However, you’re not always going to have the luxury of being in position, and sometimes you’re going to be facing a bet from your opponent. Let’s say you’re facing a bet on the turn with a hand that has 20% equity. 20% of the time, you’ll make the best hand on the river and you’ll win the whole pot. 80% of the time, you won’t have the best hand, and your opponent will win the pot. You can call to try and realize that equity on the river, or you can fold.

The calculations for working out whether or not we should call are essentially the pot odds calculation – is our win % higher than the price we’re being laid, if so, call, if not, fold. However, it’s not always clear how much equity our hand has, especially when we’re thinking in terms of ranges. Sometimes you have to make estimations of what your opponent’s range looks like to determine whether or not a call is profitable.

Range analysis is an inexact science, as you’re never 100% certain which hands are in your opponent’s range. Therefore, you should pay close attention to how your opponents play to get as much information as possible to inform your decision-making.

## Combinatorics and Hand Range Analysis

- The role of combinatorics in assessing hand ranges
- Counting possible combinations of hands based on the board
- Narrowing down opponents’ likely holdings through hand range analysis

While range analysis will never be exact, you can use tools such as combinatorics to narrow down the potential hands in your opponent’s range. Combinatorics looks at the individual combinations of hands that your opponent can have in their range and uses information such as your hole cards and the board to eliminate hands from their range.

In Texas Hold’em, each paired hand has six unique combinations, and each unpaired hand has sixteen unique combinations – four suited and twelve offsuit. This means that there is a total of 1326 unique hand combinations possible in the game. Just so you know for certain that this is the case, let’s take a look at each combination.

**Pocket Pair** – AcAs, AcAh, AcAd, AsAh, AsAd, AhAd

**Unpaired Hands** – AsKs, AhKh, AcKc, AdKd, AsKh, AsKc, AsKd, AcKs, AcKh, AcKd, AhKs, AhKc, AhKd, AdKs, AdKc, AdKh

However, by using the cards available to us in our hand and on the board, we can reduce the number of hand combinations in our opponent’s range and get a more precise understanding of how their range looks. For example, let’s say our opponent has a 4betting range of AA, KK, and QQ only. We have KcKs and we call the 4bet.

Immediately, we know that our opponent can only have one combination of KK, as we have the Kc and Ks in our hand. This means that it’s impossible for our opponent to have KcKs, KcKh, KcKd, KsKh, or KsKd. If our opponent has KK, they can only have KhKd. So, this means it’s much more likely that our opponent will have AA or QQ.

We see a flop, and it comes Ac6s5h, and our opponent shoves. We know that they will check with their KK combinations, bluff with their QQ combinations, and value bet with their AA combinations. Seems like we have a 0EV play with our KK right? Wrong!

We can use the information on the board to get a more precise picture of our opponent’s range. The Ac is on the flop, which means that it’s impossible for our opponent to have AcAs, AcAh, or AcAd. This means that there are only three combinations of AA for our opponent to have, while they have all six combos of QQ.

So, facing the shove from our opponent, we know that they will be bluffing twice as often as they will be value betting, so we can profitably call with our KK.

While this is an exaggerated example, combinatorics is used by professionals even when ranges are wider. Knowing that holding

## EV and Pot Control: Navigating Bet Sizing

While pot odds and EV are separate concepts, you should be considering the two of them in tandem when choosing your bet sizing in a given situation. This is because the bet sizing you choose will have a direct effect on the pot odds you’re giving your opponent, which will increase or decrease the EV of their call, and simultaneously, your bet.

Not every situation is going to yield a +EV bet; sometimes, no matter what you bet, you’re going to be losing money in the long run. This is often the case when you have an extremely weak hand, or if your opponent has an extremely strong range. In these situations, it makes much more sense to keep the pot as small as possible. This means that you get to realize your equity for as little as possible, and allows you to ramp up the action once you’ve hit the card you need.

It’s also perfectly possible to make our opponent’s call +EV or -EV, depending on the size we bet. Let’s say we’re looking to bet the flop with a strong top pair hand in a $100 pot. We know that our opponent may have a better hand, but we can still get value from the weaker parts of their range, and we can fold out hands that have considerable equity against us if we bet.

If we bet small, we manage to keep in more of our opponent’s weaker hands, but we also make it profitable to call with higher equity hands. However, by making a larger bet, we can reduce the EV of our opponent calling with strong equity hands like straight draws and flush draws, reducing the overall EV of their range. Plus, if our opponent does call with these hands, we’ve forced them into an equity mistake, as they won’t win the hand often enough to justify a call.

## Poker Math in Action: Case Studies and Examples

We’ve talked a lot about hypothetical situations so far, but let’s take a look at some real-world examples where poker math has been used by some of the best professional players to win a huge pot.

### Dwan Coolers Ivey

In the Full Tilt Poker Million Dollar Cash Game back in 2010, Tom Dwan coolered Phil Ivey in an over $1 million pot when he made a higher straight against the lower straight of Phil Ivey. Let’s take a look at the pivotal decision in that hand and see how poker math helped Dwan make his decision.

The hand starts three-handed with Dwan raising from the button to $6000 in a $1K/$2K cash game with 7h6h. Ivey comes in for the 3bet to $23,000, and Dwan makes the call. The pair are over $500,000 deep, and they see a flop of Jc3d4c, giving both players a straight draw. Ivey bets $35,000 into a $49,500 pot, and this is where the big decision point is for Dwan – should he make the call on the flop?

Well, let’s look at his odds. He has to call $35,000 to win $49,500, so he’s getting 2.4:1 on a call, which means that he has to win the hand 29% of the time to win. Does he have the right price to call? Well, he has four outs to make his straight on the turn, so multiplying his 4 outs by 2 (using the 2/4 rule we learned earlier), he’ll know that he has 8% to win. Even if he could guarantee that he would see a river, he doesn’t have the right price to call.

However, the pair are sitting extremely deep, and given that Ivey 3bet preflop and bet aggressively on the flop, Dwan will likely assume that Ivey has a strong hand and that he could make a lot of money if he hit his 5 on the turn; in fact, he may be able to get everything that Ivey has in front of him.

Let’s run the implied odds calculation and see how much Dwan needs to make on later streets to make this call profitable.

**Implied Odds = ((1/0.08) * $35,000) – ($84,500 + $35,000)**

**Implied Odds = (12.5 * $35,000) – $119,500**

**Implied Odds = $437,500 – $119,500 = $318,000**

So, Dwan needs to make another $318,000 on the turn and river to make this profitable. Given how strong he must have thought Ivey was, and given the fact that they had over $500,000 behind, Dwan must have thought this was possible and made the call.

Dwan then hit the miracle 5 on the river to cooler Ivey and win a pot of over $1 million.

### Handz Bluffs Into Keating

In one of the biggest pots in US televised poker history, Handz lost a $1.15 million dollar pot on the HCL show after bluffing into Keating’s flush.

The hand starts with Handz raising to $9,000 on the button with As7h and getting three calls, including Keating in the $1,600 straddle. The flop comes Ts6s5d, giving Keating a flush draw, and Handz some back door draws. The action checks to Handz who c-bets for $25,000, the other two players fold, and Keating calls.

The 4d on the turn was a complete brick, but after Keating checks again, Handz fires another bet for $70,000. Keating quickly makes the call and is rewarded with the 7s on the river. He leads out for $155,000, and Handz decides to shove for $310,000 more. Was this shove a good decision?

Well, as Handz holds the As, he knows that it’s impossible for Keating to have the nut flush, so by using combinatorics, he makes it much more likely that he would get a fold than if he made this bluff with a random hand. To determine how profitable this bet was, we need to consider Keating’s leading range on the river.

He’s known to be a very loose-aggressive player, but after check/calling two streets, a lot of his river leads are going to be flushes. We can add a decent percentage of random pairs that he may turn into a bluff or other missed draws that made it to the river, but it’s reasonable to assume that around 70% of the time, he’s going to have a flush.

Now, the next thing we need to decide is how many of those flushes Keating will call with. He’s a much looser player than anyone else at the table, so it’s reasonable that he would talk himself into a call with around half his flushes in this situation. Given that we think he’ll call around 50% of his value range, we can estimate that he will call with 35% of his leading range.

By using the bluff EV equation, we can work out the EV of this bluff by Handz.

**EV = (0.65 * $536,000) – (0.35 * $310,000)**

**EV = $348,400 – $108,500 = $239,900**

So, as you can see, the EV of this play from Handz is over a quarter of a million dollars, he just got unlucky that he ran into the second nuts.

However, what would this EV calculation look like if we think Keating calls all his flushes?

**EV = (0.3 * $536,000) – (0.7 * $310,000)**

**EV = $160,800 – $217,000 = -$56,200**

See how quickly things can change if you adjust your opponent’s calling ranges? This is why paying attention to how your opponent plays is extremely important, even if you’re using poker math, as it allows you to better understand and predict how they will react in certain situations, which allows you to make more accurate calculations.

## Conclusion

Although it might seem intimidating at first glance, the mathematics of poker isn’t something that should scare you. The calculations are easy to do once you understand what they’re doing, and with a bit of practice, you’ll soon be able to perform these calculations in your head. Once you’ve mastered the math of poker, you’ll find that you are able to make more precise decisions and thoroughly understand the profitability of each play. You no longer have to guess whether or not you’re making the right decision – with poker math, you’ll know for certain.

## FAQs About Poker Math

### How can I overcome my fear of math and start using it effectively in poker?

The most important thing to remember is that the math isn’t as complicated as it may seem – there are no quadratic equations or linear algebra to learn, it’s all basic math that can be easily learned with practice.

### Are there any recommended resources for learning and improving poker math skills?

Aside from this article, there are a number of poker math workbooks that offer examples of the math we’ve shown here that you can work on at home.

### Can poker math be applied to all poker variants, including Texas Hold’em and Omaha?

While not all of the math we’ve covered can be used in every game, some aspects of poker math are applicable across multiple variants of poker. Things like pot odds and expected value don’t change, no matter what game you’re playing.

### How do I incorporate poker math into my decision-making process without slowing down the game?

The easier answer is to practice! Mental math can be hard for some players to perform quickly, especially if they’re learning it for the first time. The only way to be able to quickly use poker math in-game is by practicing it until it becomes second nature.

### Are there any shortcuts or mental tricks for quickly calculating odds and probabilities?

Yes, the 2/4 rule is an easy shortcut that allows you to work out the probability of you hitting your outs across one or two cards. To work out how often you’ll hit your outs on the next card, simply multiply your number of outs by two, and you’ll get the percentage of the time you’ll hit what you need. To work out the same answer across the turn and river, simply multiply your number of outs by four!

### How do I balance the use of poker math with reading opponents and exploiting their tendencies?

In poker, you should be using all the data points available to make the most informed decision possible. While understanding poker math is essential to becoming a winning player, you shouldn’t use it at the expense of other aspects of the game, such as understanding how your opponents play. For example, if you have a solid grasp of how your opponents play, you can adjust your range to increase the EV of certain plays.

### Can poker math help me become a more disciplined and profitable player over the long run?

Yes, one of the reasons that players make mistakes is because they don’t understand why it’s a mistake. They call off with flush draws in situations where they’re not getting the right odds, but they don’t even know what odds are! If you know that you’re not getting the right odds to call with a certain hand, it becomes much easier to fold and wait for a better spot.

### Are there professional players who attribute their success to a deep understanding of poker math?

All winning players in the modern game have a deep understanding of poker math, and wouldn’t be winning players without it.

### What is the difference between using math for decision-making and relying on intuition?

If you rely on intuition, you’re likely going to be staring down your opponent to decide whether or not they have it before making your call. If you use poker math, you’ll use your knowledge of poker ranges to estimate the EV of your call and determine whether or not it is profitable.

### How has the integration of poker math evolved over time, and what role does it play in modern poker strategy?

While the math in poker has been the same since it was invented, it is used far more often now than it ever was. In the old days, even the good players didn’t use poker math much, instead relying on their hand-reading skills to determine what their opponents had. In modern poker, an understanding of poker math is essential if you want to be a winning player.