We've all heard that poker is a numbers game, so it make sense that there is a correlation between poker and math, right? but do you really know how important those numbers are? As a preamble to this in-depth discussion, it is worth pointing out the raison d'être of playing poker.

Poker is a betting game that requires players to use tactics and strategy to compile the highest-ranking hand.

Typically, the player with the strongest hand wins the pot. Throughout the course of a poker session, various elements come into play, notably poker psychology, poker rules, and poker statistics. A multifaceted approach to the game has to be implemented, in order for players to succeed at the highest levels. 

The math behind poker is fascinating to say the least. We're going to take you behind-the-scenes into the inner mechanics of the game, where rigorous mathematical analyses reign supreme. While poker is an easy game to learn, it is a challenging game to master. Many of the finest poker players – those whose names feature prominently in the Poker Hall of Fame - are whiz kids with poker math.

Our in-house poker experts have picked their brains for some nuggets of advice about learning poker math, and understanding what all these numbers really mean. While this is an introductory article to poker math, some of the concepts are difficult to understand, so feel free to reference this article as many times as necessary until everything comes into sharp focus.

What is Poker Math and Why is it Important?

Viewed from a rudimentary perspective, it is clear that poker is all about odds and probabilities. Poker math is focused on the ever-changing odds in a game, as actions are taken, and cards are dealt. Those who understand the mathematics of poker are better poised for victory. All skilled poker players have a firm grasp of the following concepts:

•    Pot odds
•    Fold equity
•    Poker equity
•    Implied odds
•    Bluffing odds
•    Expected value
•    Poker hand combinations
•    Probability of specific cards being dealt

The utility value of each of these concepts differs from really important to moderately important. Once you're ready to start learning poker math, the game takes on an entirely new meaning. Players with an understanding of poker odds are better positioned for success at the poker tables.

It may come as somewhat of a surprise that many casual poker players have an aversion to math. Even some of the most experienced professionals readily attest to using their gut feel to play and win poker games rather than crunching the numbers. While there are certainly outliers on the spectrum, the vast majority of successful poker players carefully scrutinize the aforementioned concepts.

While the stats that we are about to introduce are certainly complex, most poker players already use simple math to play their games. Since many of us are instinctual poker players (we base our plays on how we feel), it's nice to know that poker math can be used to validate our actions, or to discredit them and cause us to rethink our decisions.

If there is a highly unlikely chance of a specific action paying dividends, it is probably best not to act. All of this will come into focus as we progress through this material. As an online poker player, it's important to understand each of the terms listed above so that you can make the best possible betting decisions.

What are pot odds?

Simply put, pot odds evaluate the proportion of the next bet relative to the size of the poker pot.  As an example, let's assume that the pot is $900 and a player must call a $100 bet to continue in the hand. At this time, the player is receiving 9:1 pot odds. If the player calls, the size of the pot is $1000, indicating that the $100 bet comprises 10% of the pot size. This is important insofar as it determines whether a player will call a bet. If the odds of improving your hand are less than the pot odds, it's worth calling the bet. If the pot odds are less than the odds of improving your hand, it's worth folding.

What is fold equity?

If you believe that your poker opponent will fold when you bet, you have additional equity in the form of fold equity. This happens all the time in poker. Whenever you are betting, there is a chance that your opponent will fold. As we all know, this is the easiest way to win a poker pot. This probability (that our opponent will fold) increases our overall equity in the hand, because it boosts our chances of winning the entire pot.

The equation for determining overall equity is as follows: Hand Equity + Fold Equity = Total Equity. It's not an absolute science, since we are working with unknown variables. However, we can have substantial fold equity if it is highly likely that our opponent will fold to our bet. We can have minimal fold equity if it is unlikely that our opponent will fold to our bet. 

What is poker equity?  

Poker equity refers to the percentage of the pot that is yours for the taking based on your odds of winning. If the numbers suggest that your hand has a 65% chance of winning, it stands to reason that you have 65% equity in the poker hand.

The surest way to calculate poker equity is to use a poker odds calculator. Once you plug in the cards you are holding, and the cards your opponent is holding, the poker odds calculator will spit out a few percentages. If the pot size is $1000, and you have a 55% chance of winning, that translates into $550 poker equity.

Poker equity percentages do not guarantee winnings, because multiple variables come into play at any given time. They are based on statistical analyses with data collected over thousands of hands. The odds calculator generates poker equity figures based on the expected outcome.

What are implied odds?

Much like pot odds, implied odds will help you to determine whether it is worth your while calling if you are facing a raise. The implied odds will tell you how much you can expect to win after you hit your draw. You have good implied odds if your expectations of winning more money increase after you make your draw, and you have poorer implied odds if you expect not to make any more money from your opponent on future rounds of poker.

Implied odds are presented as an estimate based on your understanding of your opponent and the cards on the table. As a case in point, you may have been dealt a 4 & 5, and the Flop comes out to 3, 6, and Jack. Clearly, you have a shot at an open-ended straight draw. If your opponent bets, you have great implied odds because you can make a straight. However, if the board (The Flop, Turn, and River) is favorable to all players, your implied odds are no longer favorable to you and your opponent will likely fold.

What are bluffing odds?

The inimitable bluff is one of the most popular actions in poker, although it's also the most overused and least understood by novices. Bluffing is an art that requires tremendous experience to successfully pull it off. It is a form of ‘psychological intervention’ designed to throw your opponents off their game.

If you are playing for $10,000 and you are about to bet $200 to win the pot, you might make this bet even if your bluffing odds are just 10%. 10 x $200 equals $2000, which is substantially less than the $10,000 up for grabs. If you lose, you lose $200. If you win, you win $10,000. Calculating your odds of success with bluffing can be difficult, given the complexity of each situation and the abilities of your opponents in question.

What is the expected value?  Expected value (EV) is a mathematical concept whereby a player ‘expects’ certain outcomes given the probabilities of those outcomes. For example, your odds of correctly guessing red or black with playing cards are 50-50, like a coin toss.

If somebody offered you $10 for every correct guess and a zero-dollar penalty for every incorrect guess, you would have an expected value of $5 on every card that is dealt with you. Of course, you will never get paid $5, it is a theoretical concept since you either get paid $10 or $0. However, when the probability of an event occurring has a percentage associated with it, those odds must be multiplied by the potential payout to get the expected value.

What are poker hand combination odds?

This is a broad topic which can be broken down into multiple stages, including the pre-flop stage, and the post-flop stage. In the pre-flop stage, you can expect the following probabilities of winning (based on the following card combinations):

•    Two High Cards (and your opponent has a low pair) – 55%
•    Middle Pair (and your opponent has one high card and one low card) – 71%
•    High pair (and your opponent is holding two low cards) – 83%

During the post-flop stage, things look a little different, with the following probabilities that your cards will make valuable poker hand combinations:

•    Your pair will Flop a set – 12%
•    You'll hit a Flush with two suited cards – 6.50%
•    Your pair will flop 4 of a Kind – 0.25%

What is the probability of specific cards being dealt?

This is a little easier to calculate, given that we understand how many cards there are in a deck, and how many cards there are per suit. If you want to know the odds of a face card being dealt from a standard 52-card deck, simply calculate the number of face cards in a deck and then divide it by the total number of cards in the deck. FYI: Aces do not count as face cards.

There are Jacks, Queens, and Kings in every suit. That means there are 12 royals per 52 card deck = 23.08% chance of a royal being dealt. For those who are going after a Flush, it's relatively easy to calculate the odds of hearts, spades, diamonds, or clubs being dealt, given that there are 4 suits with 13 cards per suit in a standard deck. 

*While you are assessing the numbers, you may come across figures in a ratio format, or in a percentage format. It's much easier to look at percentages and compare them than it is to look at different sized ratios and compare them.

Learning Poker Math: Frequently Asked Questions (FAQs)

Whether you’re playing Texas Hold'em poker, Omaha Hi-Lo, or 7 Card Stud, you are going to come up against formidable opponents as poker tournaments reach the showdown. At this stage of the game, you will likely have mastered poker math and understand a myriad of terms related to odds and probabilities, outs, and card distributions, et al. We've compiled a short list of frequently asked questions for your benefit:

Q - When are the odds in your favor in poker?

A - Always evaluate the odds being offered. This is a skill that takes tremendous practice, insight, and understanding. If you are prepared to bet $100 in a $200 pot, your opponent needs to call your $100 bet to make it a $300 pot. This changes the odds of the game, and requires players to think about things like implied odds, and the prospect of making a profitable call. If your win rate is better than the odds being offered, you're likely to win more than you are likely to lose.

Q – What are poker outs?

A- Every time you play poker, there are specific cards which will improve the strength of your hand. These are known as outs. If you're looking to make a Flush and you have 2 hearts in your hand, you need 3 hearts from the Flop, Turn and River to make it happen. Given that there are 13 hearts in a 52-card deck, and just 11 of them potentially remaining, you will need to assess your chances based on what you see on the table. If the Flop reveals one spade, one heart, and one club, then there are just 10 possible outs from 47 cards on the Flop. That’s 1:4.7 of getting a heart on the Turn card. 

Q – If you understand poker odds will you always win?

A - No. Poker math does not guarantee that you will win every hand that you play. It is your best tool to become a successful player, but there are many other factors at play. One of them is variance. It can be challenging to master poker odds, but eventually you will understand things like implied odds, pot odds, expected value, probabilities of hand combinations and so forth.

Q – Are heuristics as relevant as math in poker?

A – This in-depth article has touched on many statistical concepts such as implied odds, pot odds, variance, poker outs and the like. However, scant attention has been given to heuristics. This concept deals with anything that is used for the purposes of problem-solving through experimentation, particularly through experiential learning.

While math is an important component of poker, it doesn't always translate into tangible results. Things like three betting are incredibly important in the game. So are continuation bets. The nonmathematical elements of poker such as emotional stability, risk aversion, and skills contribute substantially to the win rate of top-class players. If you can avoid going on tilt, your game is automatically elevated.

Now that you know about poker math, it’s time to put your understanding of these concepts to good use.

About the Author

With digital marketing strategies in his blood, Louis Wheeler has traveled around the world, exploring gambling cultures and gaining experience in casino games from 2003. If you are in a casino anywhere around the planet, you may find him right next to you, playing blackjack, roulette or texas hold'em. 

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