Have you every felt overwhelmed by the numbers in odds charts? When seeing the winning percentages, probabilities, and various other mathematical probabilities in poker games, have you ever wondered where they come from?
In this article, we will discuss some common probability principles in Texas Hold'em poker games and how they are used.
Don't worry about the poker math. Everything we cover in this article can be easily understood, allowing you to better understand how to calculate a hand’s probability before the flop.
Starting from some basics
The two numbers we use most often in calculating hand probability are how many cards are in a deck and how many cards we want to be dealt in the game. For example, when we want to be dealt one specific card:
Remember there are 52 cards in total in a deck:
- If we want to be dealt a 9♦，the probability would be 1/52, because there is one 9♦ in the whole deck.
- On the other hand, if we want to be dealt an Ace, regardless of its suit, the probability would be 4/52, because there are four Aces among the 52 cards.
- The same logic also applies to the situation where we only need one card from a certain playing card suit, diamond♦ for example. Then the probability of dealing any ♦ would be 13/52, because there are 13 cards of the same suit.
To sum up, the probability of dealing any random card is 1/52, the probability of hitting any specific card such as Ace or King, Queen is 4/52, and the probability of dealing any suit is 13/52.
Now, let’s level up a bit
Suppose you are holding a 9♦, and you want to be dealt a pair. Now you are wondering what is the probability that you are dealt a 9 again.
- We’ve already known that the probability of being dealt a 9 is 4/52, which is the one you are holding now.
- And the probability that you are dealt another 9, regardless of its suit, would be 3/51.
Have you noticed that the numerator and the denominator of hitting the second card changed? When we got the first card, there were only 51 of 52 cards left. And when we hit the first 9, there are only 3 other 9s left in the whole deck.
These numbers will keep changing as we move along in the game. Always pay attention to adjust the numbers based on the information you learn.
Tips for calculating probability
- When we use the word "and", we will use multiplication for the calculation.For example, we want to be dealt a King and a Queen.
- When we use the word "or", we would use addition for calculation.For example, we want to be dealt a 9♦ or a 9♣.
Moving on, we will talk about how to work them out in detail.
Probability of being dealt two specific cards
Multiplication
Suppose we want to know the probability of being dealt a King♦ and a Queen♣, (pay attention to the “and” we use here).
- Probability of being dealt a King♦ first would be 1/52.
- Probability of being dealt a Queen♣ would then be 1/51.
So the probability of hitting a specific poker hand, one king and one queen, is 1/2652.
P = (1/52)*(1/51)
P = 1/2652
So, the probability of being dealt any combination of hands in poker is 1/2652. However, this number represents that you get a King♦ first and then a Queen♣. But do you really care about their order? Not really.
In other words, there are actually two ways to hit this hand ( get a King♦ first or get a Queen♣ first). Therefore, the probability that we are dealt this hand should be multiplied by 2.
P = 1/2652*2
P = 1/1326
To sum up, there are 1,326 possibilities for any starting hand we hit in the Texas Hold'em poker game.
Poker probability for a certain hand
What if you only need a specific hand and don’t care about their suits?
Again, we use multiplication.
Taking KQ as an example:
- The probability of hitting any King, regardless of its suit, would be 4/52.
- Probability of hitting any Queen, regardless of its suit, would be 4/51 (Because one King has already been dealt and there are only 51 cards remaining).
P = (4/52)*(4/51)
P = 16/2652 = 1/166
Same as before, what is calculated here is actually the probability of getting a King first and then a Queen. Of course, the probability is the same if you do it vice versa, so just multiply it by 2.
P = 16/2652 * 2
P = 32/2652
P = 1/83
As a result, the probability of being dealt any hand type in poker is 1/83, regardless of the poker hand ranking.
Poker probability for a certain range
What if you only need a KQ or a QJ? Pay attention to the”or” in the sentence.
What you need to do now is to calculate all the probabilities of hands you want and add them together
Since we have discussed before, when seeing “or”, we will use addition for calculation, therefore:
- The probability of being dealt KQ is 4/52 * 4/51 = 4/663
- The probability of being dealt QJ is 4/52 * 4/51 = 4/663
Hence,
P = 4/663 + 4/663
P = 8/663
If you want to add more hand types into your range, simply add up their probabilities like we have done here. Pay attention when you want to calculate the probability of a pair, for example AA. The probability of being dealt a pair is slightly different from KQ.
- The probability of being dealt AA is 4/52 * 3/51 = 1/221
It should be noted that these arithmetic problems are not for you to sort out when you sit at the poker table. They are the basis for you to review and organize your Texas Hold'em poker strategy in your spare time. You need to know these principles in order to learn and practice more advanced strategies. Keep learning and gradually you will become more and more experienced and confident.
Poker Hand Probability Overview
What we covered today are some of the most common concepts and principles that will help you work out the probabilities of being dealt various hands before the flop. What’s more important is that you try and use this knowledge in your games.
You may feel that this article does not actually affect your skills or style of play, but the underlying logic of skills and style of play is supported by these basic mathematical concepts.
I hope this article can serve as a stepping stone to more advanced study, helping you to more efficiently absorb and learn more advanced and complex poker theories.
Keep learning, keep practicing and start building up your own strategy. Otherwise these are merely abstract concepts which cannot help you win.