Mathematics and Probabilities in Poker

Contrary to what many believe, poker is more about mathematics and probabilities than luck or chance. The critical element of winning in poker is to use available information, calculate it, and make the correct decision.

This article will walk you through the basic concept of probability and its relevance in online poker games. We will also go through probabilities of being dealt specific hands, basic pot odds, and their likelihood of winning against other combinations.

What is a Combination in Poker?

In poker, a combination refers to a specific instance of a particular hand. It can also refer to how a poker hand forms. Combinations in poker are essential for making strategic decisions and determining the strength of a hand. It allows players to weigh their options when deciding to fold, commit to a hand, or make strategic bets.

Combinations at poker include various hand types such as pocket pairs, straights, flushes, and royal flush. Players often use combinations to assess the strength of their hands and make informed decisions during the game. They help players understand the value of their hands and consider the possibilities of their opponent's hands. 

What is Probability?

Probability is the branch of mathematics that deals with the likelihood of occurrence of a particular outcome when an event occurs. Example: When you flip a coin, there are two possible outcomes: heads or tails. In this scenario, flipping a coin is an event. Heads and Tails are two possible outcomes. Hence, the probability of one specific outcome, say Heads, is calculated as follows.

p(Heads) = One event/Two Possible Outcomes = 1/2 = 0.5 or 50%

Must Read: Importance of Making Notes on Players

Probability and Poker Hand

When using a deck of cards, the possible outcomes are much greater than flipping a single coin. A standard deck consists of 52 cards,  categorized into 4 suits (Club, Diamond, Heart, and Spade), and each suit has 13 ranks (Ace, King, Queen, Jack, and 10 to 2).

So, the probability of getting dealt an Ace as the first card is calculated as:

p(Ace 1) = Number Aces in the deck/Total Number of cards in the Deck 
= 4/52 = 1/13 = 0.077= 7.7%

Similarly, the probability of getting dealt a card of one specific suit is calculated as:

p(Club) = Number of clubs in the deck/Total Number of cards in the deck = 13/52 = 1/4= 0.25 =25%

As the cards are dealt, the number of total cards remaining in the deck decreases, changing the probability of getting the second card. So, after getting Ace as the first card, receiving Ace as the second card would be:

p(Ace 2) = 3/51 = 1/17 = 0.059 = 5.9%

Now if we want to find out the probability of getting dealt pocket Aces, we need to multiply both:

p(AA)= (4/52) * (3/51) = (1/13) * (1/17) = 1/221 = 0.00452 = 0.45%

Statistically, this means you will get a pair of aces once every 221 Hands.

Also, the probability of getting dealt any pocket pair will be 5.9%. So statistically, you will be dealt any pocket pair once every 16 hands you play.

Apart from Pocket Pairs, poker hands are categorized in various categories.

The first card will be the higher ranked of the two cards dealt along with the suit, and the second card will be the lower ranked along with the suit. So if you are dealt an Ace of Diamonds and 2 of Spade, we will address it as Ad 2s.

A = Ace
K= King
Q= Queen
J= Jack
T= Ten
2-9 as it is.

Club= c
Diamond= d
Hearts= h
Spade= s

• Suited Hands
  Suited hands mean that both the cards dealt to you are of the same suit. For example, Kh Qh or As Ks.
• Pocket Pairs
  The same ranked cards dealt to you are called Pocket Pairs.
• Offsuit Hands
  Off-suit hands are the hands that are not pocket pairs and are of different suits. For example, Td 9s or Ac Qh
• Suited Connectors
  Suited connectors are the hands which are of the same suit and are in continuation. Like Ts 9s, Jd Td, 6c 5c, Th 9h.

Let's also understand how many ways a poker hand can be dealt to you.

The formula of 2 cards being dealt is
Hand Combos = (n*(n-1))/2
Where n is the number of available cards.

So the total number of possible hand combinations through this formula will be:

Total Combos = (52*(52-1))/2 = 1326
Because there are 52 cards in the deck

Combinations of Pocket pairs will be:

Pocket Pair Combos = (4*(4-1))/2 = 12/2 = 6
Because there are four cards of the same rank in the deck

As there are 13 different ranks, the total number of pocket pair combos

Total Combos of Pocket Pairs = 13*6 = 78

Combination of Suited Hand

Suited cards Combo = (13*(13-1)/2 = 156/2 = 78
Because there are 13 cards of the same suit in the deck

As there are 4 different suits, the total number of suited hand combos

Total Combos of Suited Cards = 78 * 4 = 312

The total number of off-suit combos = 1326 – 78 (Pairs) – 312 (Suited Cards) =936

Must Read: Understanding Positions in Poker

Summary

You have now understood the probabilities calculations for getting specific hands and also the combinatorics of poker starting hands. Poker probabilities play a significant role in poker strategy, as players use mathematical calculations to improve their chances of winning.

FAQs

What is the order of combinations in poker?
The order of combinations in poker, from the highest to the lowest, is Royal Flush, Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pair, One Pair and High Card.

What is the best combination in poker?
Royal Flush is the best combination in poker. It consists of A, K, Q, J, 10 of the same suit. The Royal Flush is the best Straight Flush one can get in poker.

What combinations are priority in poker?
The Royal Flush poker combination is followed by Straight Flush, Four of a Kind, Full House, Flush, Straight, Three of a Kind, Two Pair, One Pair, and High Card, in descending order of priority.