Mathematical Strategies to Winning No-Limit Texas Hold’em shows players how a changing deck alters familiar odds and how to turn those calculations into decisions at the table. What first led you to question the ideal probability model?

I have watched several videos and read many books which analyzes poker probability, and felt that the stated probabilities seemed much higher than what would be expected, from a mathematical viewpoint. Upon reverse engineering several of these probabilities, I have concluded that the ideal model always assumes that the round is played with ten players and that every ‘out’ is still expected to be in the deck. This is a best-case scenario and rarely occurs in real life. As more pocket hands are dealt, the probability that a desired ‘out’ may be dealt to another player or serve as a burn card increases, removing the probability of it appearing on ‘the board’. But, if you assume that these ‘outs’ are still in the deck, there is an increased probability that they will appear on ‘the board’ due to the smaller number of cards remaining in the deck. On the other hand, for only a few players, the probability that an ‘out’ still remains in the deck is higher, but the probability of it appearing on ‘the board’ is reduced due to a larger number of cards remaining in the deck. This is the effect of transient analysis on the remaining deck. These modified probabilities help the player assess the strength of their own hand as well as that of an opponent’s, thereby allowing them to make better decisions at the table.

How did you decide which depleted-deck calculations belonged in the quick-reference tables and which readers needed to work through in detail?​

The book is written as both a math thesis as well as a playing guide. All of the equations are discussed to assist the reader in understanding the reasoning behind the calculations, and to make rough approximations at the table as needed. But they are generally way too complicated to compute during a game. Therefore, they have all been automatically computed and presented in table form for easy reference. These tables are not intended for the player to memorize immense data, but to get a general idea of the probabilities that an ‘out’ will appear on the flop, turn, or river based upon the number of dealt hands. As most players prefer to play the players rather than the cards, the quick reference tables are easily referenced to help the player make the best decision.

When your advanced model suggests a different play from conventional outs calculations, how should players weigh opponent behavior and position?​

Poker is not played by mathematical probabilities alone, but through a variety of math and psychological factors. If a player is acting as if they hold a superior hand, they either truly do or are bluffing their way to win the pot. In this scenario, you can examine the board in an attempt to predict which ‘outs’ may be in an opponent’s pocket hand and make your call, fold, or raise decision based upon probabilities that those ‘outs’ are still expected to be within the deck or dealt to the player. Position dictates when it is your turn to act. The value of your pocket hand and your position is the first important decision a player must make. Marginal hands are best played in late position and conservatively, while the best hands are generally played aggressively. This tactic, however, may be altered throughout the game to catch the other players off-guard.

Which assumptions in your advanced model are most sensitive to missing information, and how might players adapt as opponents’ ranges become clearer?​

Players generally make big calls and bets when they believe that they hold the superior hand (or wish to portray that image in a bluff). Obviously, the betting ranges are not stated specifically, as that is dependent upon each player’s risk tolerance. Although the tables of pre-calculated probabilities can aid each player in determining the probabilities of his hand winning as well as that of his opponent(s), he must make a risk/reward decision during each betting round.

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