Why you should have a math approach to poker - WPT Global

Poker used to be a game where psychology and being able to read your opponent were the most important things you needed to become a winning player. Over the past decade or two, the game has become a lot more math-centric, as some very smart people have been able to figure out the underlying math that exists in poker. The main benefit of playing in a mathematically-sound approach is that it helps you play a consistently better game. Give your math a try and download WPT Global today.

Knowing the math of poker, and always being able to rely on it means that you have a solid foundation for your game. Having to rely purely on psychology and reading your opponent opens you up to making mistakes when you're unable to do so properly. We'll break down some of the most common calculations you can make in poker so you can try out a math-based approach for yourself.

EV (Expected Value) Calculation in Poker

One of the most useful things you can do when you have a math-based approach to poker is that you can figure out the EV of any play you make. For the uninitiated, EV stands for Expected Value, and is simply the amount of money you can expect to win or lose on average for each action you take. There is a mathematical formula you can use to work out the EV of a bet or a call at the poker table:

EV = (Win% * $Win) - (Lose% * $Lost)

Simply put, you multiply how often you expect to win by the amount you would win, and subtract the multiple of how often you expect to lose by the amount you would lose. Let's look at a hand and put some real numbers into this equation to make it easier to understand:

You're in a $1/$2 cash game and you've gotten to the river with Queen-high after missing your flush draw on the turn. You don't think you can win at showdown, so you want to bluff. You think you can get your opponent to fold 70% of the time if you bet $100 into a $100 pot, but you want to check if this is a +EV play. This is where you can use our equation:

EV = (75% * $100) - (25% * $100)

EV = ($75) - ($25)

EV = $50

You can see that if we're right in our assumption that our opponent will fold 75% of the time, the EV of our bet will be $50. However, we're never going to know for certain what the exact percentage of hands our opponent will fold, so it's worth adding a buffer of 10-20% in your EV calculation to account for this.

Without knowing this equation and understanding the math behind it, we'd have no idea whether any play we make is profitable or not. We'd have to wait thousands of hands to see if we're profitable in the games, and even then we wouldn’t know for certain! Learning equations like this one can help you quantify exactly how profitable, or unprofitable, a certain action will be.

Break-Even Calculation in Poker

Another thing you can do in poker if you have the right knowledge is work out exactly how often a particular play (bet/call/raise) needs to work in order for it to break-even. This is exactly what the break-even calculation will allow you to do, and once you know your break-even point, you can compare this to how often you think your play will work to get a rough idea of whether or not it will be profitable. Let's look at how the calculation works:

Break-even = Bet Size / (Bet Size + Current Pot)

The break-even percentage is the proportion of the bet size compared to the amount we stand to win. Let's say you're on the river with a missed straight draw and you want to bet half-pot as a bluff, but you want to figure out your break-even point to see if it will be profitable. To do this, we simply plug in the numbers to the above equation:

Break-even = $50 / ($50 + $100) = $50/$150 = 0.3333 = 33.33%

We can see that if we were to bet half-pot on the river as a bluff, we would need it to work 33.33% of the time to be a break-even play. If we didn't have this understanding of the math, we'd be completely in the dark when it comes to whether or not our play would be profitable. We'd have to resort to guessing - which is also best to avoid when playing for money.

Frequencies Calculation in Poker

When playing poker at a decent level, it's important to try and stay as balanced as possible in your play. To be balanced, you must have a proportionate amount of value-bets and bluffs in your betting and checking ranges and be calling/folding at the correct frequencies.

To be able to not be exploited by your opponent's bet, you need to reach the minimum defense frequency (MDF) - that is the minimum percentage of your range that you need to call for your opponent's bet to not be automatically profitable. This is the formula to calculate the MDF when facing a bet:

MDF = (pot size / (pot size + bet size)) * 100

Let's look at a real-world example:

We're on the river in a $1/$2 cash game, and our opponent has just bet $50 into a $100 pot. How much of our range do we need to call to prevent our opponent from automatically profiting? To figure that out, we just need to plug the numbers into the formula:

MDF = (100 / (100 + 50)) * 100

MDF = (0.6667) * 100

MDF = 66.67%

This means we need to call at least 33.33% of our range to stop our opponent profiting by over-bluffing this spot.

Not only can we use this to determine how often we need to call to stop ourselves being unexploited, we can also use it to determine if we can automatically profit from a bet. If we're the ones betting, we can use this MDF calculation to work out how often our opponent should defend, and if we think they'll fold more often than this we can make an exploitative over-bluff and increase our profit.

The History of Math in Poker

While math has come to the forefront of poker over the last couple of decades, the history of math in poker goes a lot further back than that. The original game of poker only used 20 cards, and players were simply dealt a 5-card hand that they would bet on whether or not they thought they had the best one. Knowing the likelihood of being able to make certain hands would allow you to make a lot of money in those earlier games.

The game then moved to a standard 52-card deck and the "draw" was added to the old 5-card game. Even more so than the last iteration, knowing how likely it is to make a certain hand and using that information to decide what you should draw to would give you a massive advantage over your opponents.

Even as early as 1875 there were books being released on the math of poker. Henry T. Winterblossom's "The Game of Draw Poker, Mathematically Illustrated", is one of the first poker strategy books. You can see from the title just how much emphasis was placed on the math, even in the early days. As the years went by and new poker games were being created, more and more of these strategy books were released - all having an emphasis on the mathematical side of the game. In the bible of poker, Doyle Brunson's Super System, Mike Caro breaks down the math of all the popular poker games at the time, from 5-card draw to 7-card stud, to Texas Hold'em. 

Moving into the 21st century, players started to look at the math of poker in a different way, particularly when it came to tournaments. Players by this point already knew about pot odds and the chance of hitting their draw, but there wasn't a lot of information on the mathematical concepts of optimal tournament play.

In the 2003 book "Harrington on Hold'em", Dan Harrington and Bill Robertie popularized the "M" concept (or M-ratio), which reflects how shallow or deep a player's stack is compared to the blinds and antes. Since then, lots of advanced tournament math has made its way into the forefront of the game, particularly ICM (the Independent Chip Model), which assigns a monetary value to each player's stack based on the tournament payout table which can be used to make +$EV decisions in the later stages of tournaments.

Fast forward to today, it's almost impossible to be a good poker player without having a very solid understanding of math. A lot more advanced concepts have made their way into the game, like EV calculations, combinatorics, pot geometry, and many more. It turns out your teacher was right - it pays to learn math!

Luck vs. Skill in Poker

There's long been a debate between poker players and non-poker players about the role of skill and luck in poker. There are people that insist poker is purely a game of luck, and there are people that insist it's purely a game of skill. The answer really lies somewhere in the middle. There's no question that poker is a game of skill, but there are definitely elements of luck in the game. It doesn't matter how good you are, there's no controlling what cards come next!

Having a solid understanding of the math behind poker not only increases your skill level in the game, but it allows you to quantify the role that luck played in any particular hand/session. One of the hardest things to do in poker is to work out whether you're playing badly or just getting unlucky. By having a foundational knowledge of poker math, you are better equipped to analyze your hands to see if you're just getting unlucky, or if you have leaks in your game. Most players don't realize what they do wrong when they play, so being able to see when you're losing because of a lack of skill rather than bad luck will give you a huge advantage over your opponents in the long run.

Summary

Poker math is one of those things that you can no longer avoid if you want to be a winning poker player. Players are getting better all the time, and the way to stay ahead of them is to have a fundamental understanding of how the game of poker works from a mathematical level. If you don't have that solid foundation, it's like you're building your strategy on quicksand - eventually it will all come crumbling down.