Profitable poker requires a solid foundation in mathematics. It’s not just about (pop-culture) show-time antics when real money is on the line. Serious players understand mathematical concepts and can combine those skills with personality, tells and intimidation tactics at the table. There are four aspects to which mathematics will be needed when playing and preparing for a poker career.

1. Probability: Poker involves a lot of probability calculations, such as the likelihood of being dealt certain cards, the odds of making a particular hand, and the probability of your opponent holding a specific hand. Understanding probability can help players make more informed decisions about when to bet, call, or fold.

For example, let’s say you’re playing Texas Hold’em, and you’re dealt two cards, also known as “hole cards.” One of your hole cards is the Ace of hearts, and the other is the King of hearts. The flop results in the four of hearts, 5 of hearts, and 10 of clubs. This would cause you to ponder your odds of getting another heart on the turn (where a single card is turned face up).

Well, There are 5 known cards now and 47 unknown cards. There are 13 hearts in a deck, you have two of them, and there are two showing on the flop. This leaves you with a mathematical fraction of 9/47. (ie. there are 9 hearts left in the 47 unknown cards.) Therefore the probability of a heart on the turn is 9 divided into 47, which equals .19148 or 19% chance.

The turn card is an 8 of spades. So, that didn’t work, but you have the river card still to come. Now the odds of getting another heart are 9/46, which equals .19565 or 19.6% chance.

These types of scenarios come up all the time in poker and many players know these probabilities without the use of a poker calculator.

2. Expected value: Expected value (EV) is a mathematical concept that helps players determine the long-term profitability of a particular decision. By calculating the EV of different actions, a player can make better decisions that will result in more profit over time.

Now let’s say you’re playing Texas Hold’em, and you’re holding a pair of 8s as your hole cards. The flop comes down as 8 of hearts, Ace of spades, and 6 of diamonds. Your opponent bets into you, and you’re trying to decide whether to call or fold.

To make the best decision, you can use the concept of expected value (EV). EV is a way to calculate the average amount of money you can expect to win or lose over the long run by making a particular decision.

To calculate the EV of calling your opponent’s bet, you need to consider the probability of winning the hand and the amount of money you can expect to win or lose if you win or lose the hand. Let’s assume that your opponent has a strong hand, such as a pair of Aces or better, and that there are no more cards to come.

The probability of winning the hand in this situation is relatively low since your opponent likely has a better hand than you. However, there is still a chance that your opponent has a weaker hand, such as a pair of 6s or a lower pair, and that your three-of-a-kind 8s is currently the best hand. Let’s assume the probability of this happening is 25%.

Now, let’s consider the potential winnings and losses. If you call and win the hand, you can expect to win the amount of your opponent’s bet. If you call and lose the hand, you will lose the amount of your own bet. Let’s say your opponent bets $50, and you also have $50 left in your stack.

The EV of calling your opponent’s bet can be calculated as follows:

EV = (probability of winning * potential winnings) – (probability of losing * potential losses) EV = (0.25 * $100) – (0.75 * $50) EV = -$12.50.

Since the EV of calling your opponent’s bet is negative, it’s not a profitable decision in the long run. Therefore, folding your hand would be the better decision in this situation. Using EV to analyze the situation, you can make more financially positive decisions for the long run – even IF any particular hand goes against you.

Bankroll Management: Managing a poker bankroll effectively requires an understanding of probability and risk management. A player needs to know how much to bet in each hand based on their bankroll size and the likelihood of winning or losing the hand.

We can translate Poker Bankroll management into a live poker game at the Commerce Casino in Los Angeles. In this real-life scenario, you have a bankroll of $10,000. You’re playing at a $2/$3 No-Limit Hold’em table, meaning the minimum buy-in is $200, and the maximum buy-in is $600.

A good strategy for these tables would be to buy in for the maximum amount of $600. This is because having a bigger stack gives you more flexibility and allows you to make more profitable decisions. But for your entire bankroll you can only lose $600 of $10,000 or 6%. If you go broke and want to rebuy into the game, your buy-in would be a higher percentage of your bankroll ($600/$9,400). Having set limits on how much you bring to the table based on your entire bankroll will help determine the limits, games and tables you choose to play cards.

Also, it’s important to note that buying in for the maximum amount doesn’t mean you should play recklessly or take unnecessary risks. You should still use proper bankroll management on a micro level, assessing each hand strategically.

If you lose a significant portion of your stack, you should also consider whether it’s appropriate to rebuy to maintain your stack size. In general, a safe rule is to have at least 20-30 buy-ins for the stakes you’re playing, which means that you should have at least a bankroll of $4,000-$6,000 for this game. Every player will tell you about horrific dry spells or losing streaks. A proper bankroll management strategy will buffer these out. However, if you’re a skilled player with a good understanding of bankroll management, you may be able to play with a slightly smaller bankroll.

The final aspect of poker mathematics is more theoretical and harder to grasp – but equally important: Game Theory is a mathematical framework that helps players analyze and optimize their strategies in complex situations. Using game theory concepts, players can make better decisions and gain an edge over their opponents.

- Understanding Nash Equilibrium: One of the most important concepts in game theory is the Nash Equillibrium (conceptualized by a famous Princeton mathmetician named John Nash), which is a state in which no player can improve their position by changing their strategy, assuming the other players’ strategies remain unchanged. By understanding the Nash equilibrium for a particular situation, a player can make a more informed decision about their own strategy and potentially exploit their opponents’ weaknesses.
- Exploiting Weaknesses in Opponents’ Strategies: Game theory can help a player identify weaknesses in their opponents’ strategies and exploit them. For example, if a player notices that an opponent always folds when facing a large bet on the turn, they can use this information to their advantage by making a large bet on the turn with a wider range of hands.
- Balancing One’s Own Strategy: Another key aspect of game theory is balancing one’s own strategy to make it difficult for opponents to exploit. By mixing up their play and using different techniques in different situations, a player can make it harder for their opponents to predict their actions and gain an edge.
- Evaluating Risk and Reward: Game theory can also help a player assess the risk and reward of a particular decision, such as whether to make a large bet or call an opponent’s all-in. By weighing the potential payoff against the potential downside, a player can make a more informed decision and potentially avoid making costly mistakes.

Overall, game theory can provide a player with a deeper understanding of the strategic dynamics of the game and help them make more informed decisions. If you watch any high-stakes cash of World Series of Poker Final tables, a lot of game theory is applied at such high stakes.

Of course, it’s important to note that game theory and all mathematics in poker is just a tool in a player’s arsenal and should be used in conjunction with other skills, such as reading opponents, analyzing the table dynamics, using poker software and maintaining proper bankroll management.