As in every card game, we deal here with a finite probability field, in which the events to be measured are occurrences of certain card combinations (three-card combinations, two-card combinations or one card) on the board, in your own hand and in your opponents’ hands (called favorable combinations). The probability of achieving a specific final card formation is calculated by counting the favorable combinations that can occur. When calculating the probabilities, the information taken into account at a specific moment in the game is given by parameters such as the number of certain cards from the community cards, the number of certain cards from the cards showing (community cards dealt plus cards from your own hand) and the number of opponents. All calculations ignore any circumstantial information like viewing opponents’ cards, removing certain cards from the deck, a particular distribution of cards in the deck before dealing or fraudulent dealing.
     Among all games of chance, Texas Hold’em Poker is highly predisposed to probability-based decisions because, by the structure and dynamics of the game, the poker odds change significantly from one gaming moment to another.

We have here the so-called long-shot odds (probabilities of events that are chronologically preceded by others, calculated by using information available at the moment before the first event), which consist of: 

– Own hand probabilities – dealing with odds for you to achieve certain final card formations, calculated at two separate moments of the game: the three community cards dealt on the table (after the flop) and the four community cards dealt on the table (after the turn);

– Opponents’ hands probabilities – dealing with odds for at least one opponent to achieve a certain final card formation, calculated in three separate moments of the game: three community cards dealt (after the flop), four community cards dealt (after the turn) and all five community cards dealt (after the river).

 We also have the immediate odds (probabilities of events that follow immediately, calculated by using information of the moment just before that event): preflop odds, turn odds, odds of improving expected formations, and so on.

 Although the long-shot odds are the most important when making a probability-based betting decision, because they give information about achieving specific final card formations (which is the technical goal of the game), the immediate odds could help in evaluating and creating an advantage during each betting dialogue. Texas Hold’em Poker consists of two parts: the human action, namely, the betting dialogue and the random card distribution that results in building valuable formations. Most players take into account the immediate odds before each betting moment, even though this information is not always too relevant for the final event, but they do so because they know their opponents refer to them, too. Therefore, by making betting decisions based on immediate odds you can create an immediate advantage for yourself (for example, you can make other opponents fold). This is the psychological element any poker game holds. In fact, if the game consisted only of card distribution and building valuable formations, it would not be so spectacular.

 The immediate odds categories are: 
– preflop odds (odds of being dealt certain 2-card combinations as pocket cards before the flop)
– flop odds (odds for certain cards or certain 2- or 3-card combinations to be contained in the flop, calculated after first card distribution and right before the flop)
– turn odds (odds of a certain card being dealt as the turn card, calculated right after the flop)
– river odds (odds of a certain card being dealt as the river card, calculated right after the turn).

All Hold'em odds are in very large number and may fill more than 100 pages. We present here just a few examples of Hold'em odds:

Preflop odds

Odds of being dealt AA
     Favorable combinations: C(4, 6). Possible combinations: C(52, 2) = 1326.
     The probability is  P = C(4, 2)/C(52, 2) = 0.452%  (220:1 as odds).
     The same probability holds for any other pair.

Odds of being dealt AA or KK
     These two events are incompatible, so their odds are added.
     Each has the probability  C(4, 2)/C(52, 2), so P = 2C(4, 2)/C(52, 2) = 0.904%  (110:1).
     The same probability holds for any other two pairs.

Odds of being dealt a pair
     We have thirteen pair types and for each type the probability is  C(4, 2)/C(52, 2) .
     Then P = 13C(4, 2)/C(52, 2) = 5.882%   (16:1).

Odds of being dealt two suited cards
     For a specific symbol, we have C(13, 2) = 78 favorable combinations. Multiplying by four (symbols), we find 312 favorable combinations. P = 4C(13, 2)/C(52, 2) = 23.529%    (3:1).

Pair against higher pair
     Assuming you are dealt a pair (except AA), we can calculate the probability for at least one opponent being dealt a pair higher than yours. We can find a unique formula that comprises all possible situations. Such a formula holds as variables:
     n = number of your opponents
     p = number of pairs (as a value type) higher than the one you hold.
      For example, if you hold 88, there are six pairs higher than yours (99, 10 10, JJ, QQ, KK, AA), so p = 6; if you hold KK, there is only one pair higher (AA) and p = 1. The formula is:

   P = 6np/1225 – C(n, 2)p(6p – 1)/230300 + C(n, 3)p(6p – 1)(6p – 2)/238360500 =
6np/1225 –  n(n – 1)p(6p – 1)/460600 + n(n – 1)(n – 2)p(6p – 1)(6p – 2)/1430163000. 

For p = 1, the formula returns the following results:

Probabilities that at least one opponent holds AA, in case you are dealt KK

A table of values for the probability (P) of one opponent (a specific one, not at least one) holding a pair higher than yours follows: 

Probabilities that one opponent holds a higher pair     

  .............................................................................................................................................................

Flop odds

We have two cards showing (your own pocket cards), 50 unknown cards and C(50, 3) = 19600  possible combinations for the flop cards.

  Odds of improving three-of-a-kind or four-of-a-kind

Let C be a card from an expected formation and c the number of your own pocket C-cards (as value). c could be 0, 1 or 2. The outs (C-cards from the 50 unknown) number 4 – c. 

The probability formulas generate the following table of values:

  Odds of improving three-of-a-kind or four-of-a-kind on the flop  

     Odds of improving a flush

Let S be a symbol and s the number of your own pocket cards holding this symbol (S-cards); s could be 0, 1 or 2. The number of outs is 13 – s.

The probability formulas generate the following table of values:

  Odds of improving a flush on the flop

..........................................................................................................................................................

Opponents’ hands probabilities – Flush odds

Let us denote by C the event: At least one opponent makes a (SSSSS) flush formation by using five cards from his or her own hand and the community cards, no matter how many from each.
     We calculate the probability P(C) in the gaming moment when only three community cards are dealt (after the flop and before the turn). The variables the probability formula depends on are:
     s'' = number of community S-cards (cards with symbol S)
     s = number of seen S-cards (from own hand and board)
     n = number of opponents
     These variables must fit the conditions:  s'', s, n are natural numbers, ,  ,  ,  .
     For a (SSSSS) flush formation to be finally achieved, a minimal initial condition is that the community cards (the three dealt) must contain a minimum one of the S-cards.

  Table of values of probability that at least one opponent will make a (SSSSS) flush formation by river 

..........................................................................................................................................................

Sources

All Hold'em odds and all the math behind the game, for all gaming situations, are covered in the book  Texas Hold'em Odds. The book holds all the formulas and algorithms involved in the probability calculus for Hold'em, along with all numerical returns comprised in tables, as well as examples of how to use and apply them. See the Books section for details.