The Strength Matrix of a Hold’em Hand
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 pF the probability of F being achieved by river by your own hand and by qF the probability that at least one opponent will achieve something higher than F, if you will achieve F (qF 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 pF(1 – qF).
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.
