Hand Matrix Poker

The Poker-hand dataset happens to be extremely imbalanced, with the first two classes representing 90% of the samples, in both the training and testing set. A classifier that learns how to classify correctly these two-classes, but completely miss-classifies the remaining classes, will still achieve 90% accuracy in predictions. These statistics were compiled by a major online poker room from an expected value page from their database of millions of real live hands. Positive expectation hands are indicated by black numbers while negative expectation hands are indicated by red numbers.

1. Hand Matrix Poker Game
2. Hand Matrix Poker Strategy
3. Poker Hand Matrix Excel

For each hand, we call the matrix the strength matrix of that hand. Each column corresponds to a type of formation (1p, 2p, 3k, s, fl, fh, 4k). Each hand has an unique associated strength matrix, whose elements are calculable manually or by software program.

Players may refer to the strength of a hand in various ways, from which the mostly used are statistical. In their view, a hand is strong depending upon how often it has won in the past, when and where it occurred. In statistical terms, they assign the quality of being weak or strong in various degrees to a hand referring to relative frequency instead of probability. All kinds of software called “poker odds calculator” and based on partial simulations help them in making this assignment. There are also some simplistic rules based on counting outs that are frequently used for evaluating the strength of a hand in terms of odds (like Two Times Rule or Four Times Rule).

The strength of a hand, 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.

We must make a clear distinction between probability and stats. While the former is the most objective way to express the strength of a hand (per the above argument), the latter is what most of the 'odds' calculators return as an indicator of a strength. See my article Returning the Odds: Partial simulations vs. compact formulas for a detailed comparison between the returns of partial simulations and the returns of the compact probability formulas with respect to the Hold’em odds.

Both stats calculators and the odds calculations based on counting outs take the hand as being the card configuration of the board (pocket cards and community board). However, if we want to get a more accurate evaluation of the strength, we must take into account an additional parameter: the number of your opponents at the moment of analysis. Thus, we must define a hand as the card configuration of the board, along with the number of your opponents.

On this account, mathematics provides the most adequate object to picture the strength of a hand as being a matrix of probabilities, as follows:

Abbreviate the types of Hold’em formations with 1p – one pair; 2p – two pairs, 3k – three of a kind; st – straight; fl – flush; fh – full house; and 4k – four of a kind.

For any type of formation F, we denote by p_F the probability of F being achieved by river by your own hand and by q_F the probability that at least one opponent will achieve something higher than F, if you will achieve F (q_F are what we call conditional probabilities). For each hand, we call the matrix the strength matrix of that hand.

Each column corresponds to a type of formation (1p, 2p, 3k, s, fl, fh, 4k). Each hand has an unique associated strength matrix, whose elements are calculable manually or by software program. Each column of the matrix is called the strength vector of that hand with respect to the respective type of formation.

Assuming we have the strength matrix of a hand we want to analyze, how will we actually interpret it? The rough rule is: The higher the p-probabilities and the lower the q-probabilities, the stronger the hand. However, if we consider the p row, it is better for the p-probabilities to be higher in the second part than in the first, as the second part corresponds to the most valuable achievements. In fact, a high value of a p-probability for only one type of formation of the second part (s, fl, fh, or 4k) may be sufficient for considering the hand strong enough for aggressive raising, as example. Having high values of the p-probabilities in the first part (for 1p, 2p, or 3k) is not a positive factor in the hand’s strength, since consequently we will have lower values in the second part, which means that the most valuable formations are unlikely to be achieved. This happens because the sum of the p-probabilities has an upper bound. The strength matrix cannot be interpreted only by the p-row. The q-probabilities are also important, as they can raise or temper the trust one may have in the corresponding p-probabilities with respect to the outcome of the decision made basing on them.

For example, if a strength vector for a type of formation shows (0.55, 0.73), one may not rely on that good p-probability of over 50%, as long as the opponents may beat him/her with a q-probability of 73%, which is relatively high. Conversely, if a strength vector shows (0.17, 0.08), although 17% is not that much for achieving that type of formation, one may consider it worth that risk, as the opponents have only an 8% chance of beating him/her.

Of course, for a complete analysis, the entire matrix (all strength vectors) should be evaluated and interpreted. That is because when a strength vector shows non-favorable probabilities, one may look for alternatives among the other types of formations and these other strengths have a cumulative effect toward that hand’s strength evaluation. This is the main advantage of this method of evaluation in front of the others.

There is also a way of aggregating the data of a strength matrix in order for the strength to be interpreted through a single value and not through 14 values, coming to a strength indicator, which is a weighted mean of the products p_F(1 – q_F).

In my book Texas Hold’em Poker Odds for Your Strategy, with Probability-Based Hand Analyses I dedicated a big chapter to the calculation and interpretation of the strength matrices, followed by probability-based analyses on concrete Hold’em hands.

There has been much talk lately about the role of mathematics in poker skills. In my opinion, there is no role in the sense that a player is not required to study mathematics to see how the game of poker can be modeled and how odds are taken into account in a probability-based strategy. This is the applied mathematician’s job—to apply theory and get practical results for the players. However, if we want to use mathematics in poker strategies, we must preserve its character of rigorousness and this means that players should at least get informed on the mathematical aspects of their gaming behaviors.

About the author

Catalin Barboianu is a Romanian mathematician and author of six books on mathematics of gambling, published in several languages, which are listed in the official bibliographies of the students of several gaming institutes and organizations around the globe. Among them, Texas Hold’em Poker Odds for Your Strategy, with Probability-Based Hand Analyses (2011), Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets (2006), and Understanding and Calculating the Odds: Probability Theory Basics and Calculus Guide for Beginners, with Applications in Games of Chance and Everyday Life (2006).

He is also editor in chief of the website http://probability.infarom.ro, an online probability guide for non-mathematicians.

It’s incredibly difficult to put your opponent on an exact hand. Therefore, most of the time, you have to think in terms of poker hand ranges. Even though you don’t have a specific idea of what cards are in your opponent’s hand, a hand range gives you something to work with.

Beginners may not have thought of this, but winning players make almost all their decisions based on poker hand ranges and knowing the different types of poker hands you might get is extremely important. Your every action changes an opponent’s idea of your hand range and vice-versa. You can have anything when cards are dealt but every fold, call or raise tells something about the range of hands you can have.

Most players fail to make the effort to figure out a hand range. However, every player who intends to be a long-term winner needs to know how to analyze and weight hand ranges.

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Poker Hand Combinations

How to use a poker range calculator? In order to be able to calculate a range of hands, the first thing you need to keep in mind is how many possible hand combinations there are for different types of hands:

So what does the table tell you?

• There are more variations of non-paired hands than pocket pairs.
  For example, if you think an opponent’s hand range is 44 and 87, it’s more likely for the opponent to have 87 (16 combinations) than 44 (six combinations).
• There are four combinations of suited non-paired hands and 12 combinations of off-suited non-paired hands. Knowing this makes it easier to calculate the likelihood of an opponent having a suited hand.

Using Hand Ranges in Poker

Here’s a scenario: you raise with AK-offsuit pre-flop and the opponent calls. You know the opponent calls in this situation 15% of the time. According to Equilab, here’s the top 15% of poker hands (although someone’s 15% can include different hands–for example, one might play 66 rather than KT-offsuit):

You have AK-offsuit, and you’d like to figure out how your hand matches up against the opponent’s hand range. So you use Equilab to calculate it and you see that AK-offsuit has 61.36% equity while the opponent has 38.64%. Sounds like a good deal for you.

Most players assume they’d do well but they have never thought their opponent’s hand range through and evaluated it against their own hand. Once you get the habit of using an equity calculator, you’ll be surprised by how much or little equity certain hands have against certain hand ranges.

Your idea of an opponent’s hand range changes with every decision the opponent makes. So let’s say, instead of calling your raise, the opponent decides to raise, which he does 3.75% of the time. Here’s the hand range:

Based on the opponent’s hand range, he’d now have ~57% equity. When he just calls, you’re still ahead of his hand range; when he re-raises, your AK-offsuit is in trouble.

Weighted Hand Ranges

But an opponent may play certain hands out of his hand range more often than others. For example, instead of re-raising 100% of the time with 99, he may only do so 50% of the time. Based on your reads, you assign more or less weight to a certain hand in a range of hands and, just like before, calculate how your hand does against it.

And it can make a big difference. For example, an opponent’s hand range is AA and QQ while you have KK. In case the opponent plays both AA and QQ 100% of the time in that situation, you might consider your chances of winning poker around 50%, but if the opponent plays AA 100% of the time and only plays QQ some other % of the time, your chances of winning take a hit. You’d be going against AA the majority of the time.

How to Calculate a Weighted Hand Range

So let’s say you have JJ pre-flop and you’re up against an all-in raise. You think the opponent could do this with AA, KK, QQ, AK, and AQ. How many hand combinations do you beat and how many beat you?

Hand combinations that beat you:

Hand combinations that you beat:

You beat 32 of your opponent’s hand combinations and the opponent beats you with 18 combinations. By using Equilab, we can see that your equity is 42.60%. If, however, the opponent only has AK and AQ 50% of the time (which is a relatively realistic scenario), here’s what happens:

Hand combinations that beat you v2:

Hand combinations that you beat v2:

And by using Equilab, we can see that our equity drops to about 36%. That’s a dramatic difference in the long run. which can make the difference between whether you should call or fold, which obviously depends on pot and bet sizes and how much money you and your opponent have left.

Learning to weight poker hand ranges is worth your while. Weighting hand ranges gives you more accurate information about your chances in different poker situations as long as you determine the hand combinations right. Like in almost every skill game in the world, the more complicated a theory is, the harder it is to execute and the more profitable it is when executed perfectly. With a little bit of work and thinking you’ll get more accurate calculations.

How to Manipulate Hand Ranges

The basic idea of manipulating hand ranges is To make strong hands look weak and the other way around. This way we can get the most out of our hands because either the opponent likes to call when we’ve got a tight range of hands – meaning we can get the money in with a strong hand – or the opponent likes to fold when we’ve got a loose range, meaning we can get the opponent to fold when we have a weak hand.

One of the biggest problems is to recognize what strong and weak play is in an opponent’s opinion. Obviously Lisa and Bart are going to read situations differently and they’re going to end up having different ideas of who’s weak and who isn’t. Another problem is to know who’s been paying attention to the game and who has other things to focus on. You have to rely on the idea that the opponent follows the game at least semi-closely and adjusts his play optimally against your range of hands. Usually it’s easy to tell who’s following the game, though.

The third problem would be to understand how an opponent reacts to the way you play. There are different ways for players to react, obviously, since otherwise all the players would play the same way. Some make logical decisions, some don’t.

All of these points must be taken into consideration when manipulating hand ranges in poker. It’s of no use to build an image for yourself if the opponent pays no attention to the game. It might be counterproductive to build an image if the opponent reacts unlike you expected. You’ll always have to consider these points when building a player image and manipulating your range.

Common Mistakes – Being Optimistic

Hand Matrix Poker Game

The biggest mistake when figuring out a hand range is to have too much optimism when making decisions. For example, always believing the best and creating hand ranges that are convenient for you is a huge mistake and misses the whole idea. It’s not just about winning; it’s also about losing the least possible. By realizing your hand range is unprofitable against an opponent’s hand range, you avoid losing money.

Giving Up

Hand Matrix Poker Strategy

Losing is a part of poker and beating most of an opponent’s range doesn’t mean you’ll win all of the pots. You’ll lose a certain percentage of them, and you may even have an extremely unlikely run of losses but such is variance.

Poker Hand Matrix Excel

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