Binomial distribution: what it is and how to do it

It is quite common for scholars to attempt to describe a particular phenomenon by studying the probability of occurrence of an event associated with it.

After all, this is exactly what happens when a Six Sigma project studies a process and attempts to estimate the likelihood of a particular type of defect occurring.

That is, the binomial distribution is a type of statistical distribution. But do you know what a statistical distribution is? Do you know the difference between a continuous distribution and a discrete distribution?

If you answered "no" to any of these questions, continue reading this article and find out the answers.

What is a binomial distribution?

Theoretically, the binomial distribution is the distribution of probability of the number of successes resulting from a certain sequence of attempts, which follow the following characteristics:

• Finite sample space
• Only two possible outcomes (success or failure) for each attempt
• All elements must have equal possibilities of occurrence
• Events must be independent of each other.

When to use binomial distribution?

When starting a Six Sigma project, the Green or Black Belt should check what type of data (continuous or discrete) he&rsquos dealing with at the exit of the process. This will determine which tools will be used in project development.

It is up to this professional to define which of the numerous statistical distributions is the one that best represents the process being studied. The statistical distributions can be divided into two large groups:

• Discrete Distribution (Attributes)
• Continuous Distribution (Variable).

Discrete distributions must be used to model situations where the output of interest can only assume integer values (discrete) such as a number of faces or crowns, 0 or 1 for failure or success, or 0,1,2, 3, ... as the number of occurrences of a particular event of interest for example.

The discrete distribution can still be divided into two families:

• Binomial Distribution
• Poisson distribution.

In this post, we will address the binomial distributions. This should be used to model situations where for a given output of interest the probability of occurrences of a 'p' success and a 'q' failure is always constant.

This condition works well when batch sizes are large or in continuous productions for example.

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How to calculate the binomial distribution?

The probability of having k successes in an event following the binomial distribution is calculated by the following equation:

Where the probability of success is given by 'P', and that of failure is given by 'Q', satisfying the relation Q = 1 - P. 'x' is the number of successes in a sample, 'n' is the total number of trials.

It is worth remembering that is the combination of n values taken from k to k.

Binomial distribution: example

Suppose that incandescent lamps are manufactured in a production line. And they are packaged so that each package contains 10 units of bulbs.

A Green Belt knows that the probability of a lamp having a defect after leaving the production line is 5%. And he wants to calculate the probability that the same package contains 3 units of defective bulbs.

To help this professional, you, as a probability and statistical expert, will apply the following equation.

With the presented data we can identify that:

k = 3

n = 10

P = 0.05

Q = 1 - P = 0.95

It is very important to note that we use the probability of success at 'P', and this should not be confused with the likelihood that the lamp will not be defective. But rather the probability of the event we are focused on. That is, it is the probability of a defect occurring.

Applying these values and concepts in the presented equation, we have:

The Green Belt could then conclude that the probability of a box with 3 defective bulbs is 1.05%.

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