Dionysus.Biz

by Paul F. Watson

September 2019

Currently Under Construction

Brief Discussion of 13 Term Limit for Binomials

Comments About Binomial Equation Limitation to 13 Objects: This section is under construction, but I feel I owe the reader a brief explanation until I can finish my work on Binomial Limitations. The limit to 13 terms is really a practical limit. It does not come from math theory.

Binomial Theory: Mathematics allows the Binomial to be used regardless of the number of items; but, there are two practial problems with using the Binomial Distribution for large numbers of objects:

  1. The first problem is that a student (and everyone else) finds it tedious to use a calculator to 'crank out' answers for 20 or 50 binomial terms. Do you really want to do that much work? There are ways to cut down on the work. For example, if you just need one or two terms, something called the 'Binomial Equation' can be used to make exactly those terms without creating the entire Pascal's triangle. But often, a real world problem requires many terms of the Binomial and you are back to lots and lots of work. For large binomial models, there are alternatives. Often, you can get acceptable answers using other approaches with far less work.
  2. The second problem is more complicated. We use either calculators or computers to compute numbers from all those binomial terms back in chapter 3. I ask you two questions:
  3. There are two aspects to this question. First, regarding your calculator: Can it handle a number 10^99 power? What happens when the number gets larger than this? Similarly, what if the number gets very small, like, .000000000000000000000000000000000318? Can it handle that? Also, when you push your calculator hard with very large or very small numbers, will the '318' part of the answer suffer? So we have several questions regarding the ability of calculators and computers to handle extreme numbers.

    When you are analyzing 4 cannon shells, the 1st binomial term is 1* p^4 * q^0 (p is probability of hitting the target)

    When you are analyzing 21 cannon shells fired togethr, the 1st term is 1 * p^21 * q^0

    The probability of a cannon shell hitting its target is a small number -- likely about 10%. Don't you suppose the 1st term will get really, really small (when you compute .1 to 21st power)? If we had 50 cannon shells, the first term would become tiny. Can you calculator handle it? This example illustrates my real concern. i.e. Is there enough precision and ability to handle very large and small numbers to compute binomial values with 50 or 70 terms? With 13 (and less) terms, I have no concerns at all; but, I worry when we have large numbers of terms, representing large numbers of cannon shells, dice or kittens.

    During future improvements of St. Paul's Statistics Introduction, I will use several computer languages and variable types to find out what we can handle. I will likely describe what I have done and share my computer testing programs as well.

    For now, I will close by saying I have very practical concerns about using the Binomial Distribution for very large sample sizes. I have seen examples by published authors where they have analyzed samples of up to 50 articles; but, I encourage you to exercise caution. For now, the recommended limit of 13 terms is very conservative and safe. It is also realistic in preventing you from spending all day cracking out numbers of questionable accuracy.