Texas Hold'em Poker
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As in every card game, in Hold'em Poker we deal 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
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 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:

The own hand odds – 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);

The opponents’ hands odds – 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. This is something that distinguishes Hold’em Poker from games like blackjack, where odds are only informative for the very next move.

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 hundreds of pages. We present here just a few examples of Texas Hold'em odds:

Preflop odds

Odds of being dealt AA
The probability of being dealt AA is 0.452%. The same probability holds for any other pair.

Odds of being dealt AA or KK

The probability of being dealt AA or KK is 0.904%. The same probability holds for any other two pairs.

Odds of being dealt KK, QQ or JJ

The probability of being dealt KK, QQ or JJ is 1.357%. The same probability holds for any other three pairs.

Odds of being dealt a pair
The probability of being dealt a pair (any) is 5.882%.

Odds of being dealt AK suited
The probability of being dealt AK suited is 0.301%. The same probability holds for any other two suited cards.

Odds of being dealt A and less than J suited
The probability of being dealt A and less than J suited is 2.714%.

Odds of being dealt A and less than J offsuit
The probability of being dealt A and less than J offsuit is 8.144%.

Odds of being dealt two suited cards
The probability of being dealt two suited cards is 23.529%.

Odds of being dealt A or pair
The probability of being dealt A or pair is 20.361%.

Odds of being dealt two cards higher than J
The probability of being dealt two cards higher than J is 3.619%.

Odds of being dealt suited connectors
The probability of being dealt suited connectors (23, 34, ..., KQ) is 3.318%.

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All previous probabilities also hold for the hand of any of your opponents, from a neutral perspective (as long as we do not take into account your own hole cards).

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 –  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

 Opponents (n) 1 2 3 4 5 6 7 8 9 P (%) 0.489 0.979 1.462 1.946 2.427 2.906 3.383 3.912 4.33

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

 Your own pair 22 33 44 55 66 77 88 99 T T JJ QQ KK P (%) 5.877 5.387 4.897 4.408 3.918 3.428 2.938 2.448 1.959 1.469 0.979 0.489

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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 to one pair, three of a kind, or four of a kind on the flop

Let C be a card (as a repeated value) from the expected formation and let c be the number of your own hole C-cards. The next table notes the probabilities for your hand to improve to one pair, three of a kind, or four of a kind, on the flop.

 c One expected card Two expected cards Three expected cards 0 21.122% 1.408% 0.020% 1 16.545% 0.719% 0.005% 2 11.510% 0.244% 0

Odds of improving to a flush partially or totally  on the flop

Let S be a symbol (hearts, diamond, clubs, or spades) and let s be the number of your own hole S-cards. The next table notes the probabilities for your hand to improve to a flush partially or totally on the flop.

 s One expected card Two expected cards Three expected cards 0 44.173% 14.724% 1.459% 1 43.040% 12.795% 1.122% 2 41.586% 10.943% 0.841%

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The opponents’ hands odds – Flush odds in flop stage

For a specific opponent and a symbol S, let us denote by A' the event: "That opponent will achieve a flush (SSSSS) as his/her final hand" and by B' the event: "At least one opponent will achieve a flush (SSSSS)  as his/her final hand". We calculated the probabilities P(A') and P(B') at the moment when 3 community cards were dealt. The variables the probabilities depend on are:

s'' = number of viewed community S-cards
s = number of viewed S-cards
n = number of your opponent

The next table notes the probabilities P(B') for all possible values of these variables (the probabilities P(A') are on the row n=1).

Table of values for the probability of at least one opponent achieving a flush (SSSSS)

 s''=0 s''=1 s''=2 s''=3 s 1 2 3 2 3 4 3 4 5 n 1 0 0.277% 0.184% 0.117% 3.515% 2.606% 1.860% 19.409% 16.048% 12.895% 2 0 0.546% 0.365% 0.233% 6.359% 4.777% 3.454% 30.402% 23.533% 20.998% 3 0 0.806% 0.542% 0.294% 8.595% 6.894% 4.439% 37.557% 32.861% 26.517% 4 0 1.059% 0.714% 0.459% 10.373% 8.053% 6.890% 41.889% 37.139% 32.376% 5 0 1.303% 0.882% 0.569% 12.041% 9.350% 7.057% 45.721% 40.553% 35.631% 6 0 1.539% 1.046% 0.677% 13.610% 10.579% 7.945% 49.129% 43.599% 37.987% 7 0 1.767% 1.206% 0.783% 15.081% 11.738% 8.819% 52.117% 46.276% 40.314% 8 0 1.987% 1.362% 0.887% 16.455% 12.871% 9.678% 54.685% 48.903% 42.610% 9 0 2.200% 1.493% 0.990% 17.832% 13.794% 10.524% 57.514% 50.894% 44.874%
Example:
Community cards: (♣♣♥)
6 opponents
Let us see what is the probability for at least one opponent to achieve a flush (♣♣♣♣♣).
We have S=♣, s''=2, s=3, and n=6.
Searching in the table, we find that P(B') = 10.579%.

The opponents’ hands odds – Three of a kind odds in turn stage

For a specific opponent, let us denote by A'' the event: "That opponent will achieve three of a kind (TTTxy) as his/her final hand" and by B'' the event: "At least one opponent will achieve three of a kind (TTTxy) as his/her final hand". We calculated the probabilities P(A'') and P(B'') at the moment when 4 community cards were dealt. The variables the probabilities depend on are:

t'' = number of viewed community T-cards
t = number of viewed T-cards
n = number of your opponent

The next table notes the probabilities P(B'') for all possible values of these variables (the probabilities P(A'') are on the row n=1).

Table of values for the probability of at least one opponent achieving three of a kind (TTTxy)

 t''=0 t''=1 t''=2 t''=3 t''=4 t 0 1 2 1 2 3 2 3 4 n 1 0.026% 0.006% 0 0.660% 0.193% 0 12.657% 6.521% 0 1 2 0.052% 0.013% 0 1.295% 0.386% 0 20.579% 10.869% 0 1 3 0.079% 0.019% 0 1.903% 0.579% 0 28.115% 15.217% 0 1 4 0.105% 0.026% 0 2.485% 0.772% 0 35.265% 19.565% 0 1 5 0.131% 0.032% 0 3.041% 0.966% 0 42.028% 23.913% 0 1 6 0.158% 0.039% 0 3.570% 1.159% 0 48.405% 28.260% 0 1 7 0.184% 0.046% 0 4.073% 1.352% 0 54.396% 32.608% 0 1 8 0.210% 0.052% 0 4.549% 1.545% 0 60% 36.956% 0 1 9 0.237% 0.059% 0 5% 1.739% 0 65.217% 41.304% 0 1
Example:
Community cards: (37QQ)
3 opponents
Let us see what is the probability for at least one opponent to achieve three of a kind (QQQxy).
We have T=Q, t''=2, t=3, and n=3. Searching in the table, we find that P(B'') = 15.217%.

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The above probabilities belong to specific card formations, what we call simple events. However, a player also needs the probabilities of complex events (like events of type higher than or one or the other). The chapter Operating with and Weighting the Odds of the book Texas Hold’em Poker Odds for Your Strategy, with Probability-Based Hand Analyses deals with the rules of estimating and evaluating the probabilities of complex gaming events in order to use them in a strategy, by operating with the odds of the simple events. When we are allowed to add together the partial probabilities of a union of events for an overall probability and when we are not, methods of approximation and tips of avoiding hard calculations, are all explained in that chapter.

The strength of a Hold'em hand and the probability-based hand analysis

Any poker strategy consists of rules and if-then-else algorithms, which are based, among other criteria, on what players call the strength of a hand at various moments of the game. This strength, even though quantified in an intermediate moment of the game, is directly related to the final moment of the game, which is in the future. That is because we take the strength of a hand as an indicator of how good that hand is now in order to win at the end. Therefore we can refer to the strength of a hand only in terms of mathematical probability, the only scientific tool we have for measuring the possibility of a future event occurring, since the strength-of-a-hand concept involves measurability and predictability. The strength matrix of a hand is the main object of the mathematical model of the strength of a Hold’em hand and an important tool for weighting the odds within a probability-based strategy. Every poker strategy should take into account probabilities as objective information, however this information will be used through subjective criteria based on the personal threshold of afforded risk. Thus, any hand analysis should be based on probabilities.
Click the left button below for an overview on the strength matrix of a hand and the strength indicator of a hand. Click the right button below for some probability-based analyses of concrete Hold'em hands.

Hold'em Poker Mathematics online course

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 Author The author of this page is Catalin Barboianu (PhD). Catalin is a games mathematician and problem gambling researcher, science writer and consultant for the mathematical aspects of gambling for the gaming industry and problem-gambling institutions. Profiles:   Linkedin   Google Scholar   Researchgate
 Check this
 Gambling-math online course with Hold'em Poker 8-lesson module. Registration limited. Click here . Texas Hold’em Poker Odds for Your Strategy, with Probability-Based Hand Analyses Check the Books section for details.