How to calculate poker hand probabilities

Poker is a fascinating and storied game that has, next to blackjack, become the epitome of casino card games. Knowing the math behind isn’t considered advanced knowledge pursued only by very advanced players or those wishing to trick their way to victory. Instead, it’s very much the backbone of the game.

Next to mastering bankroll management and strategy, learning the Poker probability and odds across different scenarios is a key to increasing your chances of winning. Keep reading further to learn some of the most common probabilities and odds for both traditional poker and Texas Hold’em.

What’s the Difference Between Poker Probability and Odds?

First, we should explain the difference between probability and odds, as many people don’t realize they are two different things. We will use poker hands as an example. We should also point out that these are hypothetical scenarios for learning purposes only.

We will discuss the actual probability and odds for poker further below.

The probability of poker hands is the percentage or fraction of drawing a particular hand. For example, if you get a pair 20 times out of 100 draws, the probability is 20/100 = 0.2 or 20%. To explain odds further below, we will use some math and Y for probability.

The odds of poker hands are the ratio of drawing a particular hand vs. not drawing it. If the probability is Y, then not getting that hand would be 1-Y.

For calculating the poker odds of drawing that pair used in the probability example, we would calculate as follows: 0.2/(1-0.2), where 0.2 is the probability and (1-0.2) is the chance of not drawing a pair. So, the result is 0.2/0.8 = 0.25, or 1:4. 

That means you will draw a pair once per every four times you won’t. When the odds are written as “odds against,” the first number is the chance of you not drawing vs. drawing that said pair.

Poker Hand Probabilities and Odds

Around 85% of US adults admit to having gambled in their lifetime. With gambling being so ubiquitous, chances are a significant part of the population has played poker at least once. If you’re part of the statistics, chances are you’ve played poker. 

If so, have you ever wondered about the probability of hands?

In standard poker, there are almost 2.6 million hand combinations (2,598,960 to be exact.) With so many combinations, what’s the probability of a full house or another specific hand? Let’s take a look.

Texas Holdem Probabilities and Odds

Texas Hold’em has been the most popular and rewarding poker variant in recent years. In fact, some of the biggest gambling wins have been in Texas Hold’em tournaments. 

With that in mind, knowing some odds and possibilities specific to that type of poker is essential.

Texas Hold ‘Em probability of Getting Specific Hole Cards

Knowing the chances of getting a specific hand is very important, but with 1,326 combinations, how do you figure this out? 

Read below for poker hands probability and odds to increase your chance of winning.

Flopping Odds

It’s important to know the poker probability of winning before flopping. Read below for common flopping combinations and the odds. 

Texas Holdem Pot Odds

Calculating the pot odds is critical in the game of poker. In Texas Hold’em poker, your probability of winning is significantly lower if you’re operating without this knowledge. Read below for a breakdown of pot odds.

First, let’s go over how to calculate pot odds. If the pot is $100, and your opponent bets $50, that will make the pot $150. To call, you need to put in $50. The odds become 150:50 or 3:1. Then you need to calculate the percentage.

To calculate the Texas Hold ‘em pot odds in percentage, follow the following formula:

• First, figure out the total amount of the pot, including your call. In this case it would be $150 + $50 = $200
• Divide your call by the total pot. In this case 50/200 = 0.25
• Turn into percentage by multiplying by 100% = 0.25*100% = 25%

That means that you need to win at least 25% of the time to turn a profit. Knowing that number, you will be able to make better decisions in regard to calling a bet or not.

Poker Starting Hands Odds Based on Common Match-Ups

It’s essential to know your chances of winning based on pre-flop match-ups. Take a look at a few of the most common match-ups and the chances of winning.

Conclusion

The US is home to 2,157 casinos, and many states offer poker online, so there are more than a few ways to play the game. But you must win, right?

To increase your chance of winning, you must know the poker probability and odds of many scenarios. Everything from the prospect of getting a specific hand to the possibilities of increasing the pot and your winnings. Knowledge of the game is the key to success. 

That is why we are confident that you will be one step closer to mastering and winning at traditional poker or Texas Hold’em after reading this article.

FAQ

How do you calculate probabilities in poker?

Probabilities are calculated based on the amount of possible specific hands (e.g., royal flush) divided by the total number of hands (2,958,960). 

For example, there are four royal flush combinations (because there are four suits). 

To find the probability of a royal flush, we calculate 4/2,958,960 = 0.000001539 or 0.0001539%

(Interactive Math)

Are poker hands ranked by probability?

Poker hands’ rankings go hand in hand with probability. The lower the probability that you will get that hand, the higher its ranking is. 

For example, there are only four combinations of a royal flush, while there are 36 straight flush combinations, and 624 four-of-a-kind combinations (the 2nd and 3rd highest-ranking combinations).

(Interactive Mathematics)

How many 4 of a kind hands are possible?

There are 624 possible four-of-a-kind combinations possible. 

(Interactive Mathematics)

What is the probability of getting 3 of a kind?

The probability of getting 3 of a kind is 0.021129, or 2.11%. We have 54,912 possible combinations of getting 3 of a kind. We divide that by the total number of hands (2,598,960). 

(Interactive Mathematics)

How do you increase your odds in poker?

To increase your odds of winning, you must eliminate some mistakes that many people make. For example, thinking that betting at high odds brings the most profit is a common misconception.

Some other examples of common mistakes are:

• Being overconfident — Many players think they are better than they really are just because they’ve won here and there. Instead of thinking you can’t lose, focus on gaining experience and learning from your mistakes.
• Not paying attention — In poker, the probability of winning is much higher if you pay attention to what’s going on around you. Remember, it’s all about math and probabilities.
• Playing above your bankroll — You must be comfortable with the money you might lose. However, if you punch above your weight, you will be more focused on the money than the game.
• Over or under bluffing — Poker hand chances are different every game. If you always bluff on the same hand, other players are bound to notice it. Also, don’t hesitate to bluff from time to time when you know your chances.
• Bringing emotions to the table — You have to be focused on the game, not your personal life. Getting easily irritated or distracted can only harm your game. 

(Card Player)

What are the odds of a royal flush?

The odds of drawing a royal flush are 1 in 649,739. You have four possible hands (four suits) out of the total 2,958,960 total combinations, so we calculate the following:

In the equation 4/2,958,960, both numbers are divisible by four, so we get 1/649,739. The odds against are 649,739:1.

(GG Poker)(Upswing Poker)

What is the probability that a five-card poker hand contains at least one ace?

The probability of at least one ace is 0.341. 

We have four aces in a deck. If you take those aces, you’re left with 48 cards. The total hands of these 48 cards are 1,712,304. We subtract that number from the total number of hands in a deck, 2,598,960, and get 886,656 (total combinations with an ace).

We divide the 886,656/2,598,960 and we get the 0.341.

(Socratic.org)

Sources

• • Bumc.bu.edu
  • Card Player
  • Card Player
  • GGPoker
  • Interactive Mathematics 
  • Masterclass
  • Socratic
  • Upswing Poker
  • Upswing Poker