The Ultimate Guide to Poker Math
Understanding the mathematics of poker is crucial if you want to win. The nature of the game brings a unique and specific set of fundamental poker math concepts that form the building blocks of just about every advanced concept in the game.
In this article, we’ll take an in-depth look at the mathematics of poker, including all of the different areas you need to study to achieve success at the poker table.
Poker Math Concepts
- How Does Using The Mathematics of Poker Make You A Winning Player?
- When Should You Use The Mathematics of Poker?
- Examples of Using Poker Math
- Pot Odds
- Implied Odds
- Sklansky Dollars
- Expected Value
- Fold Equity
- Equity and Drawing Hands
- Hand Combinations
How Does Using The Mathematics of Poker Make You A Winning Player?
Players that know poker math hold a vast advantage over players who don’t. The fundamental concepts of the mathematics of poker include pot odds, equity, and expected value.
More advanced concepts, like implied odds, hand combinations, and fold equity, become essential knowledge as you move up in poker stakes. The higher you go on the cash game or multi-table tournament ladder, the more you run into opponents that hold an astute understanding of all of the concepts in this guide.
To achieve success at poker, you must become one of those players. At its core, poker plays as a game of mathematics.
When Should You Use the Mathematics of Poker?
Poker math enters into the equation of just about every situation you encounter at the poker table. The most basic concepts of the mathematics of poker, like pot odds and equity, should influence your decisions on every hand. Whether it’s preflop, the flop, or the turn, you should have a keen understanding of your hand’s equity versus the range your opponent is likely playing.
Expected value stands as another crucial fundamental poker concept. When betting, going all in, or making a big call, you should always be able to evaluate the EV of whatever move you’re about to make.
Examples of Using Poker Math
Let’s take a look at a pot odds calculation in action:
Suppose the pot is $100 and your opponent bets $50, making the total pot $150. This means you are getting 150:50 on a call, which can be simplified to 3:1
From there, you will want to convert your pot odds into a percentage so you know exactly how much equity your hand needs to profitably call the bet.
Let’s go over the quick 3-step process for converting your pot odds into a percentage.
Step 1: Calculate the final pot size if you were to call.
First, you need to figure out what the pot size would be if you called the bet. In this case, the total pot is $150 and its $50 to call, so the pot would be $200 ($150 total pot + your $50 call) if you call.
We’ll refer to this number as the final pot.
Step 2: Divide the size of the call by the size of the final pot.
In this case, that comes out to 0.25 ($50 call size / $200 final pot size).
Step 3: Multiply by 100 to get the percentage.
Now, simply multiply that 0.25 by 100 to convert the decimal into a percentage. That’s 25% (0.25 * 100) in this case.
This means that, when you call, you need to win more than 25% of the time in order to profit.
The next step would be to assess whether your specific hand has at least 25% equity versus your opponent’s range.
Pot odds represent the ratio between the size of the total pot and the size of the bet facing you. Keep in mind that the size of the total pot includes the bet(s) made in the current round.
For example, if there is $2 in the pot and your opponent bets $1, your pot odds are 3 to 1. In other words, you have to pay 1/3rd of the pot in order to have a chance to win the whole pot.
Pot odds are the mathematical foundation for calling situations in poker. Without them, we wouldn’t be able to figure out which calls are profitable and which are not.
Note: You can also use a ‘:’ to separate the values, such as 3:1.
For a deeper look at how pot odds apply in several different situations, check out this Upswing Poker article.
Implied odds are the amount of money that you expect to win on later streets if you hit one of your outs. This poker math concept, in combination with pot odds, is most commonly used to help you figure out if calling a bet with a draw is worth it.
If you expect to win more money from your opponent after you hit your draw, then you have good implied odds. But if you anticipate not being able to get any more money from your opponent on future streets, then you have little or no implied odds.
It’s practically impossible to calculate pot odds precisely because it would require quantifying and weighing countless variables — every possible card, action, bet size, etc. that could occur on future streets. The best you can do is estimate using logic.
What you can calculate, however, is the minimum amount you would need to win on future streets in order to justify an otherwise-unprofitable call.
For more in-depth examples and analysis of implied odds, take a look at this article.
David Sklansky is a pioneering author and expert in the mathematics of poker.
Sklansky introduced the concept of theoretical win to the poker world with his Sklansky Dollars model, which calculates expected value based on hand equity. Sklansky Bucks are a part of the overreaching concept of the Fundamental Theorem of Poker.
Let’s take a look at a sample hand using the Sklansky Bucks model. In this hand we go all-in preflop for $100 with:
Our opponent calls, also putting $100 in the pot with:
In practice, the only three outcomes for this hand are that we win the entire $200 pot, our opponent wins the $200 pot, or we chop and each retain out original $100 bet. We’re a 75.5 percent favorite to win this hand, and even if our opponent gets lucky and we lose this pot, our play was correct.
The Sklansky Bucks model rewards us for making the right play and calculates our theoretical win based on equity. So in this example, we multiply the $200 in the pot by our 75.5 percent chance of winning, and we win $151 Sklansky bucks. We subtract our original $100 bet to get our theoretical win (or expected value) of this play to be a net gain of $51.
For more on David Sklansky and his poker math concepts, take a look at this article.
Let’s take a look at how expected value (aka EV) works at the poker table.
To take an easy example, just think of how many times you’ve had pocket aces cracked after going all-in preflop. With very exceptional cases set aside (certain rare bubble and pay jump scenarios in tournaments), would you ever have considered folding those aces in hindsight?
Of course not. Because you know that getting your money in before the flop with pocket aces is a hugely profitable play in the long term.
Being a successful poker player depends on consistently making profitable (+EV) plays, many of which are more difficult to identify than others, and putting in enough volume to overcome negative variance (instances when you make the correct, +EV play, but still lose the pot), which is inevitable.
Let’s consider an example.
EV Example: Should You Shove All-In with a Combo Draw?
Suppose you’re on the button with $200 in a $2/4 full-ring cash game. A loose opponent opens to $16 from early position, and you call with J♦ 9♦. Both blinds elect to fold, leaving you heads up. The pot is $38.
The flop comes 5♦ 10♦ 2♣, and Villain fires a $30 continuation bet. You decide to call, making the pot $98, and leaving you with $154 behind.
The turn brings the 7♠. Villain bets $50. The pot is now $148.
Calling is a reasonable option, but let’s consider the EV of an all-in shove.
Let’s assume you’re familiar with Villain’s game, and know that she’s very capable of putting on the pressure with marginal holdings. You therefore think that if you shove she might fold 66% of the time. On the other hand, if Villain calls, you will need to hit your combo draw to win the pot.
Let’s see if this play is +EV based on the assumption that when Villain calls, it will be with a hand like T9 suited for top pair, against which your draw will have 34.09% equity.
There are three possible outcomes as shown on the tree:
- Villain folds and you win $148 (her surrendered $50 plus the $98 pot).
- Villain calls and you miss your draw, which results in you losing $154 (your all-in shove).
- Villain calls and you hit your draw, which results ion you winning $252 (the $98 pot plus her $154).
Calculating the EV for the first outcome is easy:
Villain Folds: $148 x 0.66 = $97.68
Now, let’s calculate the EV when called based on these numbers (remember: when she calls, you’ll either lose $154 or win $252):
Villain Calls and You Lose: 0.6591 x -$154 = -$101.5014
Villain Calls and You Win: 0.3409 x $252 = $85.9068
EV When Called: -$16.5014
Let’s plug that number back into our tree.
Now we can assess this play.
Villain Calls: 0.33 x -$16.50 = -$5.45
Villain Folds: 0.66 x $148 = $97.68
EV of Shove: (-$5.45 + $97.68) = $92.23
Hurray! Shoving is indeed profitable.
For more on expected value and its role in the mathematics of poker, take a look at this Upswing Poker article.
Equity is defined as the amount of the pot belonging to a player based on his/her odds to win the pot. This can be expressed as a percentage. For example, in a pcoket aces vs pocket kings preflop situation, the player with aces has roughly 80 percent equity to win the hand preflop.
Poker hand equity is perhaps the most important fundamental concept to understand of all of the poker math topics we’re covering here. For preflop play and all subsequent streets, you need to know how your hand equity stacks up against your opponent’s range.
Let’s take a look at a hand vs. hand equity calculation using PokerStove, a basic but powerful equity calculator:
Hand vs. Hand Equity
Input any Texas Hold’em hand, up to ten, and see what percentage of the time each one wins. A hand’s chance of winning is known as the equity of the hand, and understanding equity is one of the most crucial basic concepts of poker.
For example, if you’re holding a pocket pair like Q♣Q♥, you might want to know how that hand stacks up against A♣K♣. Let’s take a look at how to set up that calculation using PokerStove:
Clicking on the “Player 1” button brings up a matrix of possible hand combinations you can input for that player. Clicking on any of the other “Player” buttons allows you to choose a starting Texas Hold’em hand for that player.
In this example, Player 1 holds Q♣Q♥, and Player 2 has A♣K♣. The “Board” field in the top right of the display is empty, making this a preflop equity calculation. We’ll take a look at how to add cards to the board to calculate equity with the flop and/or turn on the board later in this article.
Once you’ve inputted the hands you want to look at, choose either “Enumerate All” (which calculates all possible runouts) or “Monte Carlo” (which offers a faster calculation but chooses random runouts to save time) and click “Evaluate”. It turns out Q♣Q♥ is about a 54.1% favorite over A♣K♣ preflop.
For more on how you can use PokerStove to learn about equity calculations, check out this article, which includes a link to download PokerStove for free.
Fold equity refers to the probability of getting an opponent to fold. For example, if you think there’s a 33 percent chance an opponent will fold to a bet in a $100 pot, you have 33 percent fold equity ($33) in that pot.
If you find yourself in a situation where your opponent probably isn’t folding no matter what, you have no fold equity. In these situations, bluffs no longer work and you must adjust your strategy accordingly.
Equity and Drawing Hands
It’s critical to know the probability of completing a flush or straight draw when calculating pot odds on the flop or turn. If you flop a diamond flush draw, for example, you hold two diamonds and two more diamonds are on the board. This leaves nine diamonds left in the deck, and if one of them hits on the turn or river you’ve made your flush.
In that scenario, you have nine outs, or nine cards that can come to complete your draw. When you flop a flush draw, you have a 35% chance of making a flush on the turn or river. If your flush misses on the turn, you have a 19.6% chance of completing the flush on the river.
Flush draws and open-ended straight draws represent the two most common kinds of draws you’ll see on the flop. An open-ended straight draw leaves you with eight outs.
The probability of hitting one of those eight outs on the turn or river is 31.5%. If the turn doesn’t complete the straight, you still have a 17.4% chance of hitting the straight on the river.
There are 52 cards in a deck, 13 of each suit, and 4 of each rank. This means there are:
- 16 possible hand combinations of every unpaired hand.
- 12 hand combinations of each unpaired offsuit hand.
- 4 hand combinations of each suited hand
- 6 possible combinations of each pocket pair.
There are 1326 total combinations of all hands that can be dealt pre-flop, from Aces to 3-2 offsuit. Here’s a visual representation of each hand type’s possible combinations:
As you may have noticed, you are three times more likely to be dealt an offsuit hand than it’s suited counterpart. This is what makes suited hands so valuable.
Flushes are very hard to make and even harder to beat. Starting out with a suited hand gives you a great chance to a hand that’s tough to beat that can win a big pot.
(Additionally, suited hands realize their equity better than offsuit hands because of their ability to flop flush draws.)
If you want to quickly reference combos you can use a Hand Matrix program, such as Poker Equilab.
For more on how to evaluate and use hand combos as a weapon, check out this extended article on the topic.