Understanding the Binomial Distribution

Coauthored with Mark Fiedeldey

The binomial distribution is a discrete distribution useful for estimating the probability of success or failure when these are the only two possible outcomes. Thus, the prefix, “Bi”. Understanding the binomial distribution — how it’s used and under what conditions – is therefore a valuable analytical tool for reliability engineers.

This distribution is applicable under the following conditions:

There are only be two possible outcomes to our test … pass/fail, yes/no, etc.

The number of trials is fixed at the beginning of the test.

The trials are independent of each other (i.e the result of one test does not influence the outcome of any other test.)

The probability of each outcome is the same for each trial. In other words, all parts on test have the probability of success or failure because they are made from the same material, fabricated to the same design, etc.

The formula for the mass density of the binomial distribution is as follows:

$$ \displaystyle\large P\left(x\right)=\binom{n}{x}p\left(q\right)^{n-x}=\frac{n!}{\left(n-x\right)!}p^{x}q^{n-x} $$

Where n is the number of trials (or the number being sampled); x is the number of successes desired; p is the probability of getting a success in one trial; and q = 1 – p = the probability of getting a failure in one trial. P(x) is then the probability of obtaining exactly x number of successes. 

Let’s consider an example:

Suppose we’re planning a reliability test. We have four test specimens that we’re going to be able to use on this test. And the part is required to have a 90 percent reliability at the end of the test.

What is the probability that will have exactly two failures if our part meets the reliability goal? In other words, if our parts are actually 90 percent reliable and we run this test with four parts, what’s the probability of having two failures during this test?

So first let’s confirm that the binomial is applicable.

– There are only two outcomes. We’re only considering whether the part passes or fails.

– We have a fixed number of trials. We’ve been given four test specimens to use on this test, and that’s all we’re going to have.

– The trials will be independent. We’ll make sure that we set up the test so the outcome of one part passing or failing does not influence whether any of the other parts pass or fail.

– And all parts have the same probability of failure. We’re going into this test stating that the parts have a reliability of 90% at the end of the test.

Having verified that we’ve met the four criteria, we know that the binomial model is appropriate for use in this application.

And since we’re looking for an exact number of failures, we can use this probability mass density function.

So in this example, then we have 4 trials. Therefore n = 4. We’re looking for the probability of exactly 2 failures, therefore x = 2. The probability of success equal 90%. Therefore, p = .10 and q = .90.

Plugging these numbers into our formula, we have:

$$ \displaystyle\large P\left(2\right)=\frac{4!}{\left(4-2\right)!}0.1^{2}0.9^{\left(4-2\right)}=0.0486 $$

Therefore, the probability of experiencing exactly two failures during this test is about five percent, even though the product meets its reliability goal.

Knowing how and when to use the binomial distribution, and a few other commonly encountered distributions in reliability engineering, will prepare you to effectively handle a wide range of analytical problems and make the best decisions possible for the products and processes you work on.

Mark Fiedeldey is a reliability engineer living near Cincinnati, Ohio.