Poker mathematics

Probabilities in poker
Probabilities in poker
10 min to examine the section
Learn the mathematical probabilities of different situations at the poker table.
Nash equilibrium
Nash equilibrium
10 min to examine the section
Study Nash's push and call ranges to become a winning player in tournament poker.
Poker odds for texas holdem
Poker odds for texas holdem
18 min to examine the section
Learn to calculate your outs and pot odds fast and easily and also solve 5 simple tasks to consolidate the material.

The mathematics of poker seems to be complicated and incomprehensible for beginners. In this article, we will explain to you in detail that there is nothing complicated in it and even a person that doesn't have a mathematical mindset will be able to comprehend everything. Understanding the basic concepts and principles of poker mathematics is the key factor that distinguishes the winning professionals from amateurs. Your poker strategy cannot be completed without the skills to calculate the equity of your hand and the pot odds.

Remember, that poker is a game of mastery and mathematical probabilities, but not of luck. Every time you make a decision at the poker table, you have to be confident that you implement only "positive" actions on the distance. Only then you can earn money playing poker in a long-term perspective. Poker math with examples will become more understandable, hence further we will show you the hand with a full mathematical analysis.

When should you use poker math?

Players use poker math when they need to calculate the odds of improving their hand and to decide whether it is profitable for them to try "to catch" their outs in reference to the pot odds.

Players make a decision basing on two factors:

  • Outs  cards that will help you to improve your hand
  • Chances of the pot (Pot Odds) – a ratio of the required bet from you to the existing pot.

To understand the principles better we suggest you to look at the poker math with examples:

NL 0.50/1.00$

  • Hero (BU): K♣ 9 ♣
  • Raise: 3$
  • SB: fold
  • BB: call 2$

The pot on the pre-flop: 6,50$

  • Flop: Q ♣ 8 ♣ 2 ♥
  • BB: bet 3$

Hero: ???


The mathematics of poker in hold'em is the easiest thing to learn because players get only two cards on their hands. Let's now discuss what are outs in poker. When we calculate the number of outs, we proceed from how many cards are left in the deck and which cards improve our hand.

In the example above Hero has the second-most powerful flush-draw. He also has an overcard - a King. This means that if the King comes out on the turn or river, you'll beat the opponent if he hits only one pair on the flop. So, how many cards can improve your hand?

  • Flush  there are 13 clubs in the deck, we see 4 of them (2 on the board and 2 as your pocket cards). This means that 9 outs are still in the deck which will give you a flush.
  • King  there are 4 kings in the deck, one of them is in your hand. This means that there are three more kings that can improve your hand.

In total, we have 9 outs for flush and three outs for a king, which is 12 outs in total.

Calculate the outs fast and easily: x4 and x2 rule.

In order to find out the percentage of winning your hand, multiply the number of outs on the flop before the turn by 4 and on the turn before the river by 2. This method will fast and easily help you to estimate your chance of winning the hand.

So, we have 12 outs on the flop, if we use the x4 and x2 rule, we can calculate very fast that the probability of improving is 12 (outs) x 4 = 48%. The exact mathematical percentage is 46.7%, but the rule of x4 and x2 gives us a fairly close and fast index for calculating.

If we don't improve on the turn, our probability to get our outs on the river will be 12 (outs) x 2 = 24% (the exact mathematical percentage will be 27.3%).

Probability table for improving hand

Number of outsChances to win on the flop 
(The х4 rule)
Actual chances to win on the flopChances to win on the turn 
(The х2 rule)
Actual chances to win on the turn

The x4 and x2 rule doesn't give you the exact value equity of your hand but helps you fast and easily evaluate your approximate odds improving.

Discounted Outs

These are outs that improve not only your combination but can also make your opponent's hand even stronger. In order to determine properly which outs are not "yours" and which are good only for your opponent, you need to be sensible when estimating the opponent's range.

As an example we use the hand above: Hero with K ♣ 9 ♣ and the flop has opened with Q ♣ 8 ♣ 2 ♥

What cards could the "donk-bet" opponent have when betting in this situation?

  • The opponent on the big blind could have KQ, K8s, K2s. Therefore, sometimes even a king's exit on the turn or river may not improve our hand.
  • There are also likely to be sets of eights and twos with which he is afraid of completing the flush-draw and makes a bet himself. And in this case, the emergence of a king on the board may not help us.
  • The presence of a stronger flush-draw of A ♣ x ♣ also takes an important place.

The poker math for beginners supposes that for a profitable game you must correctly evaluate your opponents' range when you conduct the calculation of outs.

Now that you know your odds of improving the hand, you need to find out whether it's profitable for you trying to catch your outs in reference to the pot in the center of the table.

Pot odds

The pot odds are an amount of the bet in reference to the size of the pot. When we calculate the pot odds, we want to know how much money we can win and what amount we need to put in order to do it.

What are the pot odds for Hero according to the hand above?

Community pot on the flop is: 6.50$ + 3$ (your opponent's bet) = 9.50$

Hero must put 3$ more to see the turn card and to potentially win 9.50$. Basing on this the pot odds will be equal 3 to 1. In order for the call to be "positive" in this hand, Hero has to win only 1 time out of 3 (25% of cases).

Hero is obliged to make a call in this situation, because his odds of improving in this hand exceed the pot odds. This action is profitable and "positive" on the distance:

  • The odds on improving = 48% (46, 7%)
  • Required pot odds for a profitable call = 25%

Reverse pot odds

When you evaluate your odds, you will know how much and how often you'll win if you "catch" the necessary outs for your draw. The reverse odds give you an understanding of ​​how much you will lose if you collect your draw which will be weaker than the opponent's combination.

The calculations of the reverse pot odds are similar to calculations of the usual odds. Players can continue the game and with the reverse pot odds only if your real pot odds are profitable enough, but it doesn't make any sense to pull your incomplete hand with the negative odds.

In order to play the hand properly in terms of poker mathematics, you need to do the following sequence of actions:

  • To determine the power of your hand
  • To calculate the outs for improving hand 
  • To discount some of your outs if you know they will help to improve your opponent's hand
  • To calculate the pot odds
  • To make a positive decision

For newbies this procedure may seem to be complicated but with the growth of experience you will be able to perform these actions almost automatically and accurately. Basic poker mathematics will quickly turn you from a beginner into a winning professional!