The Binomial Process (1 of 2)

The following extensive tip is derived from the ModelAssist™ training tool from Vose Consulting™. Readers should consult the ModelAssist references (in the form of Mxxx) for more information.

Introduction

In the Risk Analysis tip of August 3rd, 2005 (which you can read here), we talked about the use and importance of understanding Stochastic processes, the building blocks of risk modeling and statistics. In this and the next tip, we will focus on the Binomial Process (M0442), one of the four most fundamental stochastic processes (M0247).

The Binomial Process is a random ‘counting system’ where there are n independent identical trials, each one of which has the same probability of success p, which produces s successes from those n trials. Sounds boring? Well, we’ll explain why it isn’t!

The binomial process – you find it everywhere!

Although tossing a coin is probably the most commonly used example of the binomial process, it isn’t necessarily the most practical. Here are three real world examples of the binomial process (and thus situations of where a good knowledge of the binomial process can help you answer many useful questions):

• Six Sigma/Manufacturing: One has n product, with each having a probability of p of being defective, which causes s products to be defective.
• Banking: A bank has n outstanding loans, each with a probability of p of default, resulting in s loan defaults.
• Marketing research: We send out n questionnaires, and from the past we know our response rate is p, which results in s questionnaires to be returned.

There are certainly many examples you can come up with in your own industry. So, why is it now so useful to understand the binomial process?

Three parameters & three distributions…

As you can see in the description of the binomial process and in the three examples above, there are always three parameters {n, s and p}, that between them completely describe the Binomial Process. What is very interesting is that associated with each of these three parameters there are three Crystal Ball distributions (a.k.a. assumptions) that describe the uncertainty about or variability of these parameters. These three distributions require that one has knowledge of two parameters in order to estimate the third. You can see this graphically in the (important) figure below.

Figure 1. Relationship between the three distributions of the Binomial Process

The above figure is really very useful when we pose question about any binomial process, which will go as follows:

• If we know n and p, we can estimate s with a Binomial distribution
• If we know s and p, we can estimate n with a NegBinomial distribution
• If we know n and s, we can estimate p with a Beta distribution

We will further illustrate how this works, using the three examples described above. In this week’s risk analysis tips, we will examine the first example a bit closer and in doing so learn about two Crystal Ball distributions; the Binomial Distribution and the Yes/No distribution. The other two examples (and three additional Crystal Ball distributions) will be reviewed in the next risk analysis tip.

Example 1 – Six Sigma/Quality (the Binomial and Yes/No distribution)

Question: Let’s assume one buys 250 products, with each having a probability of 0.05 of being defective. How many products will be defective?

Answer: The wrong answer here would be 250 * 0.05 = 12.5! Instead, looking at the figure above, we know that n (=250) and p (=0.05), and thus we can use the Binomial distribution to estimate the number of defective products (s). The answer therefore is a Binomial distribution with n = 250 and p = 0.05 (see below).

You can see that there is an 80% probability that the number of defective products is between 8 and 17.

Note 1: A special case of the Binomial distribution is the Yes/No (a.k.a. Bernoulli) distribution. The Yes/No distribution is equal to the Binomial distribution when you only take one sample, i.e. when n = 1. Thus Binomial (p, 1) = Yes/No (p).

Note 2: From the example above, you can see that the binomial distribution is one of the three distributions that describe the uncertainty or variability of the three parameters of the binomial process. Therefore, don’t mix up the binomial distribution with the binomial process!

Other examples of the binomial distribution: The binomial distribution has an enormous number of uses. In addition, many other stochastic processes can be usefully reduced to a binomial process to resolve problems. For example:

• Binomial process:
  • The number of dry holes when we drill n times;
  • The number of defective items in n from a production line;
  • The number of people who buy a product if n people enter your store;
• Reduced to binomial:
  • Number of machines that last longer than T hours of operation without failure;
  • Blood samples that have zero, or >0 antibodies;
  • Approximation to a Hypergeometric distribution (see M0034

Example 2 and 3

Both examples will be discussed in the next Risk Analysis Tip.

What to do next?

A great deal of risk analysis problems can be tackled with knowledge of the building blocks of risk analysis and statistics - stochastic processes. ModelAssist for Crystal Ball from Vose Consulting contains hundreds of risk analysis topics and model templates.

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* The material within this ‘Risk Analysis Tip’ comes from one of the hundreds of topics available in ModelAssist for Crystal Ball. ModelAssist for Crystal Ball gives a more detailed explanation of the above methods and any risk analysis techniques involved.