by George 'The Engineer' Epstein
With so many poker books available (including my own two rather unique books), I hesitate to write about any. But Thomas M. Green’s Texas Hold’Em Poker Textbook is very different. So is Mr. Green. I have never met the gentlemen, except through correspondence. He is a retired math professor who knows the game of poker quite well, and uses mathematical concepts to perfection.
The first part of his book teaches probability as applied to the game of poker, and how to calculate the odds. Carefully selected problems related to hold’em, with pertinent answers help to better understand the math concepts. Then he reviews the basics of combinations-andpermutations, explaining their use in poker. For example, in hold’em, what is the probability of making various hands, assuming you stay all the way to the river?
|Probabilities for Final Hands in Hold’em|
|Four-of-a-Kind: 0.17 percent||Full-House: 2.6 percent|
|Flush: 3 percent||Straight: 4.6 percent|
|Three-of-a-Kind: 4.8 percent||Two Pair: 23.5 percent|
|Pair: 43.8 percent||Less than a Pair: 17.4 percent|
Over 60 percent of final hands will be one pair or lower. Only 15 percent will be three-of-a-kind or higher, assuming you stay to the river.
Suggestion: As you read this book, if the math seems too tedious or difficult, just scan those pages and study the tables and charts at the end of each section. You can go back to the math later. Pay attention to the hundreds of problems with solutions that will teach and inform you.
Holecards. Before the flop there are 13 possible pairs, with six combinations of each, considering the suits—78 pairs (6 x 13) in all. With 1,326 possible two-card combinations, expect to be dealt a pair in the hole almost 6 percent of the time (78/1326 x 100). But, a particular pair—like A-A—will occur only 0.45 percent of the time—once out of 221 hands. About 94 percent of your hands will be non-pairs (1326 - 78 = 1248 out of 1326 hands). The probability of being dealt no ace is 85 percent, so don’t be surprised when you get none in the hole for long periods of time. With nine players at the table, you can expect at least one opponent has an ace in the hole if you don’t.
|Probabilities for Holecards|
|A Pair: 6 percent||Two Suited Cards: 23.5 percent|
|Connectors: 14.5 percent||
Unsuited, Non-Pair, Non-Connector:
The Top-22 Hands. Green provides a chart of the top 22 starting hands and their relative rankings. The probability of being dealt one of these is approximately 10 percent. So anyone staying in most of the hands dealt is playing mainly hands not among the top 22. Weaker hands can be played in some situations, especially from late position.
The Next Level of Play. Green says, “You not only have to know your chances with the first two cards, but you need to consider the possibilities (probabilities) for the cards that your opponents have.” He offers rules of thumb to estimate the probability that an opponent holds a higher pair than yours, depending on the number of opponents in the game. At a table with nine opponents, the probability that one opponent has a higher pair than your K-K is only 4 percent. With holecards of 9-9, that increases to 20 percent, and it’s 31 percent if you hold 5-5. If you are playing in a heads-up game, your chances are much more favorable: With K-K, the chance your single opponent has a higher pair (A-A) in the hole is only 0.5 percent. With 9-9, it’s 2.5 percent, and with a small pair like 5-5, the probability he holds a higher pair is only 4.5 percent.
Another interesting probability-related topic. If you don’t have an ace in the hole, what is the probability that an opponent has an ace? With four or more opponents, the probability is over 50 percent that at least one player has an ace in the hole. With 8 opponents, it’s almost 80 percent.
George “The Engineer” Epstein is the author of The Greatest Book of Poker for Winners! and Hold’em or Fold’em?—An Algorithm for Making the Key Decision and teaches poker at the Claude Pepper Sr. Citizen Center in Los Angeles. Contact George at email@example.com.