1. Introduction

Texas Hold'em is one of the most popular variants of poker, known for its blend of skill, strategy, and chance. The game, which involves betting, bluffing, and strategic decision-making, can be analyzed through various mathematical principles. This analysis explores the fundamental concepts of probability, combinatorics, expected value, game theory, and risk management in Texas Hold'em, providing insights into how these principles guide optimal play.

2. Basic Probabilities

In Texas Hold'em, each player is dealt two private cards (hole cards) while five community cards are dealt face up on the "poker table". The goal is to use any combination of the seven cards to form the best five-card hand. The foundation of strategic play in Texas Hold'em is understanding the probabilities of different hand types and situations, and then judging from your own hand what combination is needed to maximize your advantage.

2.1. Hand Probabilities

Combinatorial mathematics can be used to calculate the probability of being dealt various hands. For example, the probability of being dealt a pocket pair is:

P(Pocket Pair)=[(4C2)*(48C2)]/(52C2)=5.88% (1)

One of these is the combination of nCk, where n is the total number of items in the set and k is the number of items to be selected. In this case, n is 5 and k is 2, which means we are selecting 2 cards of the same suit from the 5 cards and 2 additional cards. This results in many possible combinations, as we are selecting from the remaining 48 cards.

2.2. Draw Probability

The probability of hitting a specific card type in the flop (the first three community cards) can also be calculated. It is also important to judge what cards you need based on your hand. After a simple estimation of the probability, it can increase the probability, for example, the probability of hitting a royal flush (three cards of the same rank) in the flop stage when holding a pocket pair (two cards of the same rank) is:

P(Set on flop)=((2C1)*(50C2))/(50C3)=11.8% (2)

This reflects the chance of one of the remaining two cards of the same rank appearing among the three flop cards.

3. Combinatorics of Hand Rankings

Texas Hold'em uses a standard ranking of poker hands, from high card to royal flush. The number of possible combinations for each hand type can be calculated to assess their rarity. This calculation not only helps players understand the relative value of different hand types, but also enables them to make more informed decisions in the game.

For example, royal flush is the strongest hand in poker, but its combination is extremely limited, usually only four possible combinations (one hand per suit). In contrast, the combinations of random hands are very large, so they are more common. By analyzing the rarity of each hand type, players can better assess the strength of their own hand and the possible hand types of their opponents, and then develop strategies accordingly to increase their chances of winning.

Royal Flush: There are 4 possible royal flushes (one for each suit) out of a total of 2,598,960 possible 5-card combinations:

P(Royal flush)=(4C1)/(52C4)=0.000154% (3)

"Full House": To calculate the possible number of full houses, we first choose three card points of the same rank (there are 13 choices), then select three card points from the remaining four cards (4C3), for pairs, we select one card point from the remaining 12 cards and two card points from the remaining four cards:

Total full house=(13C1)*(4C3)*(4C2)=3744 (4)

Thus the probability of being dealt a full house is:

P(full house)=3744/2598960=0.14% (5)

4. Expected Value

The expected value (EV) is a key concept in Texas Hold'em when making decisions. It is important to determine how many high card combinations there are that can be made with your hand based on the current board, and how much you can bet accordingly. It measures the average amount that a player can win or lose per bet, taking into account the probability of different outcomes.

The expected value is calculated as the probability of Alice winning the pot times the new pot amount, deducted by the amount she bets [1]. Note that this calculation emphasizes that as soon as Alice places a bet, she should no longer consider that money to be her's to lose, but rather part of the pot that she can win (sunk cost). The formula for EV is:

EV=(P(Win)*Amount Won) — (P(Lose)*Amount Lost) (6)

For example, suppose you have a 40% chance to win a hand and stand to win $100 if you win but lose $50 if you lose. The EV calculation would be:

EV=(0.4*100) — (0.6*50)=10 (7)

This means, on average, you expect to gain $10 per such decision.

5. Pot Odds and Implied Odds

5.1. Pot Odds

The pot odds refer to the comparison between the current size of the pot and the amount of the bet you need to call. It helps determine whether it's worth calling based on the probability of winning with the hand you have. It's crucial to assess your approximate pot odds during the game to determine whether to fold with weak hands or go all in with strong ones, while still considering the probability [1]. Put simply, pot odds means is there enough in the pot to call a bet.

If the pot is $200 and your opponent bets $50, the pot odds are:

Pot Odds=50/(50+250)=20% (8)

If your probability of winning the hand is greater than 20%, calling the bet is considered profitable.

5.2. Implied Odds

Implicit odds not only consider the current bottom, but also the future bets you can win if you complete your hand. When the current odds are not favourable to you, these implicit odds are very useful, because you hope that in the next round of betting, if you get the cards you need, you can win extra money. This consideration allows players to continue evaluating the value of their bets based on potential future benefits when faced with unsatisfactory current odds.

For example, if you are chasing a Shunzi or the same flower, and the current odds are not attractive, the calculation of the implicit odds will help you estimate the additional chips obtained from your opponent after successfully completing the hand [2]. We refer to a player's range as the hands he plays in a given situation. In general, a player's range does not change from hand to hand. That is not to say that the player should be predictable. Analysing the opponent's betting mode and possible cards, players can judge the probability that the opponent will continue betting on future rounds, to calculate the potential implicit probabilities more accurately.

In summary, implicit probabilities not only provide players with a tool to make decisions in unfavourable situations, but also encourage deeper strategic thinking and psychological tactics, making poker games more complex and interesting. Understanding the importance of implicit odds will help players make wiser decisions in the game, to improve their overall performance and win rate.

For example, if you need to call a $50 bet into a $200 pot but expect to win an additional $150 if you hit your draw, your implied odds would be:

Implied pot=200+150=350

Implied Odds=50/350+14.3% (9)

If your draw’s probability of completing is better than 14.3%, the call is justified based on implied odds.

6. Game Theory and Strategic Play

Game theory is a mathematical theory that studies the mutual influence and interdependence of decision-makers. It primarily focuses on the strategic choices made by different participants (called "players") in their decision-making processes, especially in situations of competition, cooperation, or conflict. Game theory provides a framework for analyzing and predicting the behavior of these participants in specific environments [3].

They should regard game theory as part of the body of mathematics that is used to model those entities that consistently select elements from mutually exclusive action sets, resulting in patterns of choices, which, allowing for some stochasticity and noise, can be statistically modeled as maximization of utility functions.

6.1. Nash Equilibrium

In poker, game theory often uses Nash equilibrium to describe strategies where no player can improve their expected outcome by changing their strategy unilaterally. In Texas Hold'em, Nash equilibrium strategies are particularly relevant in heads-up play and tournament settings [4]. Nash equilibrium is a set of strategies, one for each of the n players of a game, that has the property that each player's choice is his best response to the choices of the n–1 other players. It would survive an announcement test: if all players announced their strategies simultaneously, nobody would want to reconsider.

For example, in a heads-up situation, a player might adopt a mixed strategy where they bet and fold certain percentages of the time to remain unpredictable and balanced. If both players use optimal mixed strategies, neither can gain an advantage by deviating from their equilibrium strategy.

6.2. Optimal Betting Strategies

Optimal betting strategies are to balance your aggressiveness and passiveness based on your hand strength, position, and opponent's tendencies. This includes:

Balance [5]. GTO is optimal because it is unexploitable; no player can increase their expected value by altering their strategy. These strategies are also referred to as "balanced", which is paramount of GTO poker. It involves mixing up your actions — bluffing, betting for value, and checking — to prevent opponents from detecting any predictable patterns. This ensures that your play remains a mystery for adversaries.

Bluffing: Using probabilistic reasoning to decide when to bluff. If you believe your opponent will fold enough of the time (based on their tendencies and the size of the bet), a bluff may be profitable.

Value Betting: Value betting is when you believe you have better cards than your opponent and make a bet that reflects this [6]. In some ways, it is the opposite of bluffing, where you don’t believe you have good cards and are hoping your bet scares away other players.

Because of this, value betting is one of the biggest poker skills to master. It is what allows you to have a big pot with a winning hand. Otherwise, those good cards will go to waste on a pot that barely makes you anything.

7. Conclusion

Texas Hold'em is a complex game in which mathematical principles play a crucial role in decision-making. By understanding probability, expected value, odds, implied odds, and game theory, players can make more informed and strategic decisions. Mastering these concepts not only improves a player's game, but also allows them to adjust their strategy based on different situations and opponent dynamics. The interaction between mathematics and strategy makes Texas Hold'em a challenging and rewarding game. This thesis has some points in game theory that have not been deeply studied, such as the defensive value in optimal betting strategies, which can be focused on Nash equilibrium and game theory for further in-depth research.