Looking Back

It is time to sit down, relax and look back at what has been discussed in the previous 21 articles. (Have there been that many articles already?)

For the most part the articles have dealt with the probabilities of certain events taking place in various poker games. Some of the articles have dealt with hand rankings and others have dealt with the chances of certain outcomes for the board in both hold'em and Omaha. One aspect all of these articles have in common is that basic counting techniques have been employed to solve the problems. What do we mean by basic counting techniques?

One basic technique is partitioning the problem into a collection of smaller problems. Sometimes the smaller problems must be partitioned again until eventually we reach small problems which are solved easily on their own. We then combine the solutions of the small problems to obtain a solution of the original problem. We use the so-called multiplication and addition rules to combine the solutions. The multiplication rule is invoked when combining solutions for problems which are connected by the word and, whereas, the addition rule is invoked when combining solutions for problems connected by the word or.

Let's consider an example which illustrates the above points. Suppose we would like to determine the number of two-card hands which add to 10 (using the usual blackjack conventions). The two-card combinations adding to 10 are A-9 or 2-8 or 3-7 or 4-6 or 5-5. So we partition the problem into the five smaller problems of counting the numbers of each of the five types of hands. Since the connective word relating them is or, we add the respective solutions to obtain the final answer. In order to solve a typical subproblem such as the number of two-card hands of the form A-9, we break it into the problems of the number of ways to have an ace and the number of ways to have a nine noticing the connective word here is and. There are four ways to have an ace and four ways to have a nine. Since the connective word is and we multiply and obtain 16 hands of the form A-9.

The other basic idea used over and over is the notion of a k-combination of an n-set, that is, a choice of k objects from a set with n objects. For example, a five-card poker hand from an ordinary 52-card deck is a 5-combination of a 52-set. The number of k-combinations of an n-set is given by the formula n!/k!(n-k)! and appears in almost all the calculations we have done. Continuing with the preceding example, we see there are 52!/5!47! = 2,598,960 five-card poker hands in a standard 52-card deck.

I taught mathematics to thousands of university students for approximately thirty years and observed that by and large the topic of counting problems created more difficulty than almost all other topics. Comments and questions I have received from readers since starting this series of articles confirms there are many people who find counting problems difficult. Why is this so?

One reason is that counting problems typically are not discussed in the usual secondary curriculum meaning that students encounter them for the first time in university courses. This is changing to some degree because of the importance of discrete mathematics, where counting problems are first encountered, for computer science. There is now exposure to discrete mathematics in some secondary programs. Another reason is lack of practice. Very few people master mathematical topics without practice and learning to solve counting problems takes practice. A third reason is the lack of a powerful universal technique for solving counting problems. Thus, the solutions for counting problems tend to be somewhat ad-hoc and students find this disconcerting. They discover a neat trick to solve a particular problem and then never get to employ the trick again. One of the biggest problems students encounter is not clearly stating what is to be counted leading them to solutions for the wrong problem. The latter problem is closely related to the problem of not partitioning the problem to be solved into its individual constituents.

Hopefully the interested readers have considered the ways I have partitioned some of the problems considered earlier and perhaps learned something from it. However, even when you know what you are doing, there is danger lurking at all times when dealing with counting problems. First, it is easy to make arithmetic mistakes. Some of the numbers discussed in earlier articles were extremely large and this makes arithmetic mistakes . Second, when breaking problems into smaller problems it is easy to overlook one of the subproblems and forget to factor it into the final solution.

Now comes confession time. First, there is an arithmetic mistake in the article Low Board Blues: I. I stated that there are 1,037,008 boards which do not allow a low in Omaha high-low because I forgot to add the 224 boards which have five low cards where four of them form four-of-a-kind. Fortunately, this has such a small effect on the probability of a low board not occurring that the 0.399 given in the article is correct. The ironic aspect of this mistake is my comment to Bib Ladder in the subsequent article Low Board Blues: II not to overlook five low cards containing four-of-a-kind. Second, there are several problems with the probabilities given in Low Board Blues: IV and Low Board Blues: V. For some reason I do not understand, the analysis given there is not directed towards the problem clearly set forth at the beginning. Thus, the numbers have no relevance to the original problem.

Because of these mistakes and my aversion to making such stupid mistakes, I decided to start carefully documenting my calculations before I submit the articles. These are available at my website and I invite interested readers to go to my website and examine these computations. I would love to receive either confirmation from others that the computations are correct, or comments pointing out any errors. The website is http://www.math.sfu/~alspach and still is not complete.

Let me now look ahead. In my next article I am going to clarify from Low Board Blues parts IV and V. Following that, I am going to introduce a more sophisticated counting technique which is useful for certain kinds of problems.