Poker mathematics: outs, probabilities, pot odds

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Updated 15.04.2024
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Poker mathematics: outs, probabilities, pot odds

What are we talking about here? The mathematics of poker seems difficult to many new players. However, in fact, it is quite simple to master and thereby greatly increase your chances of winning.

What should we focus on? Of course, you need to start with the basics and understand what poker mathematics, outs and probabilities are in the first place and how to calculate them. It is also very important to be able to apply this knowledge in practice.

What is the role of math in poker?

Strategy in this game is based on probabilities of certain events. Understanding the psychological characteristics and habits of your opponents is also important. This information allows us to make our mathematical predictions more accurate. Poker mathematics covers a wide range of concepts, including outs, probabilities, pot odds, equity and fold equity.

Poker mathematics

It is necessary to analyze the chances of winning in the current hand. Math serves as a guide for experienced players, helping them determine optimal moves. For example, it tells you whether it is worth making a call in a given situation or whether it is better to look for other, more profitable actions.

What Outs are in Poker and how do we count those?

Counting outs is the most important part of poker math. Outs are cards in the deck that can improve your current hand, turning it into a winning one. Let’s take a look at an example. Let’s say you have AK of diamonds on a board with a diamond flush draw. How many outs do you have in this situation? What cards can improve our combination?

There are three aces and three kings that improve us to a top pair, which could be enough to win. To achieve this, you need to draw a card that matches one of the six remaining in the deck.

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In addition, any diamond card gives us flush, which is the best possible hand here (nuts). There are 13 diamond cards in the deck. Two of them are a part of our hand and two more are on the board. Therefore, there are still 9 diamond cards left that can improve our hand.

You have 15 outs total. However, when counting, it is important to take into account and exclude from the list the so-called tainted outs – cards that improve our hand, but at the same time strengthen the opponent’s hand.

Using probabilities in poker math

Understanding outs allows you to move on to using probabilities. Once a player understands how many cards in the deck can improve his hand, he can estimate how often this should happen.

Стратегия игры в покер

The following formula is used for calculations:

Probability = number of outs / number of hole cards (in the deck and in the opponents’ hands).

A standard poker deck contains 52 cards. Two of them are in our hand, and three more are on the table after the flop. This leaves 47 unknown cards. Let’s say we need any of two eights or four kings left in the deck. The probability of one of these hitting the turn is 6/47 or 12.76%.

If none of the eights or kings appear on the turn, the player still has one more chance to catch them on the river. However, it is important to remember that at this point there will be one less hole card in the deck: 6/46, or 13.04%.

The probability of getting the required card in one street is approximately 2%, and in two – about 4%.

Pot odds in poker math

What happens when mathematicians play poker? They start calculating the pot odds. The probability of getting a desired combination is determined by comparing unsuccessful outcomes, where the player does not get an out he looks for, with successful ones.

This chance can be represented in a form of a ratio. For example, 3:1 means that on average out of four attempts to get the desired card, three will be unsuccessful and one will be successful. Probabilities can also be expressed as percentages. For example, a ratio of 4:1 is 20%. When converting to percentages you just add one to a number of unsuccessful outcomes. Thus, 5.5 to 1 is 15.4% (1/6.5).

Pot odds are the ratio of the amount a player puts into the pot to the total size of the pot. In other words, this is a ratio of how much a player invests to how much he can win if he gets a desired outcome. For example, pot odds of 4:1 means that if we invest $1, we will get $4.

Generally, pot odds are taken into account when determining whether it is profitable to call or not. If the pot odds are greater than the probability of getting the desired hand, then calling is profitable.

Two ways of applying poker math

Calculating equity outside poker tables

When a player is at the table, there is simply no time for complex calculations. It is better to start learning by analyzing your hand histories after the game. The more time you spend practicing away from tables, the more confident you will be at the tables.

A decision that closes the action is one when there is no other options available after. For example, this can happen on the river when the villain goes all-in. You have only two possible options left: call or fold. One way or another, the hand is over after you pick one of the two.

Poker strategy, how it works

Let’s take a look at another example, this time preflop. Let’s assume one of the players decides to go all-in. You have a choice: give up and fold, or call and answer his bet.

It is important to note that all calculations discussed below assume that there will be no action after your decision and the pot size will remain unchanged. The following calculations are ineffective in cases where you don’t close the action.

Let’s say we are on the river, pot is already 45 bb. Villain decides to bet his remaining 36bb and goes all-in. Hero has two possible options:

  • Call 36bb bet;
  • Fold his hand.

The question is how much equity do we need to have against villain’s perceived range for calling to be a smart decision?

To proceed with the calculation we only need two numbers. The formula takes into account the following parameters:

  • AC (amount to call) – the amount of money we have to put into pot to call villain’s all-in.
  • TP (total pot after bet) – the total pot after villain’s bet. Includes all money bet on previous streets, as well as opponent’s bet on the river.
Poker strategy, how it works
  • RE (Required Equity) – the parameter we are trying to calculate. This is the minimum amount of equity against your opponent’s range for the call to be at least break even. If in reality the equity of our hand is greater than the calculated value (RE), then calling becomes a profitable decision in the long run (+EV).


RE = AC / (AC + TP)

Poker mathematics follows all the standard rules of arithmetic. Therefore, we first sum the values in parentheses and then do the division.

AC = 36 (opponent bet on the river)

TP = 45 + 36 = 81 (pot before river action + opponent’s bet)

RE = 36 / (36 + 81) = 30.8%

When calling, a player must win at least 30.8% of the time at the showdown for the call to be justified.

Mathematical calculation is the first step in analyzing such spots. When we have specific numbers, we can evaluate the equity of our cards relative to our opponent’s range and make an optimal decision.

A very common mistake is when people calculate total pot as just a sum of river pot and opponent’s bet. However, you have to include our potential call as well, so it is his bet plus the size of the pot on the river plus the amount we should put in to make a call.

Estimation of required equity at the table

You are unlikely to be able to make complex calculations, even if you only play couple of tables. Therefore, we need a simpler method even if it is not 100% correct in terms of calculations. It does exist and is known as “RE scale”.

A player has to memorize four rules, which are:

  • If before the bet there’s no money in the pot, it would take 50% equity to call (e.g. preflop shove, although even then there are 1.5bb in the pot from the blinds).
  • Against a pot size bet, we need 33% equity.
  • Half pot bet – 25% equity.
  • Quarter of a pot bet – 17% equity.

Thus, we have four key points from which we can adjust. Many poker players use bets of the same size, which will simplify the task. However, even in the case of non-standard and random bets, you can approximate the equity you need from those key points.

Let’s look at a specific example. Let’s say your opponent bets 7bb into a 30bb pot. You can quickly estimate on the go that this is slightly less than quarter of a pot (4 * 7 = 28). Therefore, you need slightly less than 17% equity to call.

If we make precise calculations, we get the following:

  • RE = AC / (AC + TP);
  • RE = 7/ (37 + 7). Remember that this is not (30 + 7), since TP represents the size of the pot after your opponent’s bet;
  • RE = 15.9%.

It turns out that our quick estimation was pretty close to the actual number.

Useful literature on poker math

One of the best books for beginner poker players is “Poker Math Made Easy” by Roy Rounder. Its only 34 pages long. We would also recommend lying your sight on “The Mathematics of Poker” by David Sklansky.

Poker strategy, how it works

In addition, it would also be beneficial for you to reak the following books: Alan N. Schoonmaker “Your Worst Poker Enemy”, Jared Tendler “The Mental Game of Poker”, Leszek Badurowicz “Mental Edge”, Roy Rounder “Poker Math Made Easy”, Matthew Janda “Applications of No-Limit Hold’em”, James Swinney, Adam Jones “Optimizing Ace King”, MMAsherdog “The Preflop Bible”, Peter Clarke “The Grinder’s Manual”.

This library would be enough to understand the basic mathematics in poker and increase your chances of success.

Poker Math FAQ

How is mathematics useful for a poker player?

Mathematics in poker is used to evaluate the profitability of specific actions. Science allows us to determine the probabilities of winning in the current hand, taking into account the starting hand and the board. It also helps you make the right decisions, taking into account the bet size and the pot size.

How to count outs correctly?

You should count outs only after the flop and turn are dealt. Many people make the mistake of performing calculations with only their hand and no board.

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Knowing outs is a critical aspect of the game. The calculation helps determine the ratio and evaluate the pot odds.

How to use math in poker?

Mathematics is a powerful tool that should be used at the right time. Combine your math knowledge with other strategies and techniques to achieve greater success and increase your profits.

Thus, poker mathematics is a set of techniques based on mathematical principles. They help you make decisions that are more accurate. By understanding and putting it into practice, you can increase your chances of winning.

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