Two of the most ordered books on my website are Catalin Barboianu‘s Texas Hold’em Odds and Mike Petriv’s Hold’em Odd(s) Book. This is amazing given the fact that the Barboianu book is self published and the book by Petriv has been out of print for a long time (you can order a spiral bound copy). It shows that there is an interest in the mathematical aspect of the game. Most people are weak in math and are not able to figure out the odds by their own but they believe that it is important to know the odds in order to make the right decision. These people are looking for a book that can help them.

Some years ago I got an email from Rumania. The author, Catalin Barboianu, told me that he is a mathematician and that he had mathematically solved every aspect of Texas Hold’em. He had seen my software Hold’em Analyzer and thought I might be interested in his work. He asked if I would be interested in using his formulas and making a deal. I found this offer rather strange for a couple of reasons. First, he probably overestimated the sales revenues of Hold’em Analyzer. Making a deal means that Barboianu wanted – quite understandably – money for his efforts. But I never sold, nor do I do it now, enough copies of the software to offer Barboianu a just compensation for his work. Second, by studying my software he should have realized that I didn’t need his formulas. What can be done by mathematics can be done by programming but the opposite isn’t always true.

This brings us to the third point. Barboianu claimed that he had solved every aspect of the game. I didn’t doubt Barboianu’s mathematical talent but even without being a math guru myself I know this couldn’t be true. Even seemingly simple card problems can often not be solved mathematically. Take Blackjack as an example. How to play your cards correctly, called basic strategy, could only be found with the help of a computer. Or take an example from Hold’em. You have three hands, let’s say A♦J♦, J♠T♦ and 5♠4♦. Which hand wins how often in a showdown? This is an apparently easy question but too complex to solve it mathematically but with the help of a computer, it’s no big deal.

I have no doubt that Barboianu offered his formulas to other poker software programmers. Obviously they all declined as I did. Not being able to find a customer for his formulas, Barboianu decided to go ahead and publish his work as a book. This way he hopes to get some revenues for his hard work.

Barboianu’s book is divided in three parts. Part one, “Own Hand Probability”, calculates the probability to make a specific hand (one pair, two pairs, three of a kind, straight, flush, full house, four of a kind and straight flush) with three or four cards on the table. For example if you have 4Q and the flop is 23Q you have a probability of 11.655% to make two pairs at the end (p. 16).

Although the book is written in halting English, this is not a big defect because the book consists mostly of formulas, tables and pseudo code sequences. I assume that the formulas given by Barboianu are ok. I couldn’t check them all because there are too many of them but the few I looked at were correct. The errors I found seemed to be typos. E. g. the likelihood to flop a full house when you have AK is 0.0918% and not 0.00091% (p. 134).

Now to the real problems of the book. It is poorly structured. In part one he calculates the probabilities in the order full house, three of a kind, two pairs, flush, straight flush and straight for no apparent reason. Similarly in part two he starts by calculating river odds, then flop odds and finally turn odds again without explaining the strange order.

The by far largest part of the book (the first two sections) is of no help for the hold’em player. First, the formulas are too difficult to calculate at the table and it’s impossible to remember the numbers because there are just too many of them. Second, most of the numbers may be of academic interest but are useless for the player because they don’t reflect real playing conditions.

Take a look at the second example I mentioned above. You hold 6K with a board of 235KA. The book tells you with five opponents there is a 52.6% chance that somebody holds a pair of aces. Well, this is only correct if your opponents hold random cards and they play their ace to the showdown. Notice that the ace fell at the river. A lot obviously depends on the betting action before the river but under real playing conditions the chance that someone just makes a pair of aces at the end is significant less than 52.6%.

The third part of the book (immediate odds) is the most useful. It contains a lot of odds you should know as a hold’em player but you can find these basic odds in most hold’em books. Another serious problem the book has is that you just get the numbers and are not told what to do with them in real play, how it should affect your playing decisions. In the introduction Barboianu writes:

“Before any betting round, you will know the odds for you, for one specific opponent and for at least one opponent to finally achieve any possible card formation. Thus, by analyzing your hand and the dealt community cards, you will first check what formation could you possibly achieve. For each of them, you will find the odds by searching the tables. Then, you will check what possible formations could the opponents achieve (by looking at the dealt community cards) and see the odds for those higher than the ones expected for your hand … By comparing these odds (own hand and opponents’), you will make your betting decisions (call, raise or fold).” (p. 5) Barboianu may be a capable mathematician but he has no idea about poker.

To sum it up: Barboianu has presented a stunning scholarly work. Without a doubt he has invested a lot of work and time writing it. Unfortunately, it is not much of a help for the average hold’em player.

Mike Petriv’s Hold’em Odd(s) Book is quite different from Barboianu’s book. Petriv is not a mathematician but a writer and stock market investor. This is a big plus for the average reader. Petriv wanted to figure out the basic odds in Hold’em for himself but found the mathematics given by academical authors much too difficult to follow. “I read about permutations and Bernoulli’s theorem and more y = a(x-2n) equations than I could handle. And after all that, I still couldn’t figure out how to calculate Hold’em probabilities for most situations. Recently, I took a hard look at figuring out the odds and everything came together. The result is Hold’em’s Odd(s) Book, the encyclopaedia on this subject. It is for the player who would truly know this aspect, of this complicated game.

The book is a simple, comprehensive, user-friendly, layman’s approach to Hold’em probabilities. There are no leaps from obscure and irrelevant theory to difficult situations.” (p. 2) The book live up to its promise. It shows you how to calculate the most important odds and probabilities in Hold’em in a clear and understandable manner. If you know the fundamental rules of arithmetic you will have no difficulties following Petriv’s explanations. If it’s that what you are looking for Petriv’s book is an excellent introductionary text. If somebody tells you he was a favorite to win this hand after the flop you can figure out for yourself if he is telling the truth or not.

However Petriv’s book has the same problem as Barboianu’s: it doesn’t show you what to do with the odds and probabilities in actual play. Even worse is that nearly all the formulas are wrong. Petriv has no idea what brackets are and what they are good for. It all starts in the chapter “Math review” right at the beginning. He explains the order of operation: “To correctly solve math problems, the convention is to do multiplication and division first as they occur, that is, take the number left of the sign and the number right of the sign and solve” (p. 11) So far, so good. Then he gives the example 8*6*4*2/6*3*2 and thinks this equals 32/3 giving the explanation “the 6s and the 2s in the top and bottom were cancelled out.” (p. 13) Obviously he thinks 8*6*4*2/6*3*2 is the same as 8*6*4*2/(6*3*2) which is not true of course. Multiplication and division have the same order of operation which means 8*6*4*2/6*3*2 is simply calculated from left to right (the correct result is 8*4*2*3*2 = 384). Using combinations you can for example calculate the number of possible flops in Hold’em. C(50,3) is 50*49*48/(3*2*1) and not 50*49*48/3*2*1 (p. 29). You will find the same type of error at least a hundred times throughout the book.

In stark contrast to Barboianu you don’t have to be a math wiz to follow Petriv’s book. Just keep in mind that most of his formulas are flawed (not his explanations).

(Tristan Steiger)

Buy the Barboianu book!

Buy the Petriv book!