Binomial Distribution

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References: Dr. Mikel J. Harry, Juran's Quality Control Handbook

Purpose

The Binomial Distribution is used to calculate the probability of r occurrences in n trials when the probability of occurrence of an event is constant for each of n independent trials of the event.

It gives us the probabilities associated with independent, repeated Bernoulli trials. In a binomial distribution the probabilities of interest are those of receiving a certain number of successes, r, in n independent trials each having only two possible outcomes and the same probability, p, of success. 

Independent - the result of one trial does not affect the result of another trial.

Repeated - conditions are the same for each trial, i.e. p and q remain constant across trials. 

Free Binomial Distribution Training.


For example, using a binomial distribution, we can determine the probability of getting 4 heads in 10 coin tosses.

How does the binomial distribution do this?

Basically, a two part process is involved. First, we have to determine the probability of one possible way the event can occur, and then determine the number of different ways the event can occur.

That is......

P(Event) = (Number of ways event can occur) * P(One occurrence)


Anatomy

Binomial DistributionBinomial Distribution

Terminology

A. Vertical axis p - Scale to measure the probability of r occurrences of an event.

B. Horizontal axis- Scale to measure the number of occurrences.

C. The probability of having exactly two successes (r = 2) in ten trials when the probability of success in each trial is equal to 0.10 is approximately 0.2. The probability of having up to two successes is given by the area under the curve from r = zero to the point r = 2.

D. Curve of the distribution for various number of trials and probabilities of success. The total area under the curve is equal to one (1)

E. Probability Function

where q = 1 - p

Binomial Distribution formulaBinomial Distribution Formula

Major Considerations

• The population size is at least 10 times the sample size.
• Applicable to discrete distributions.
• The probability of success in each trial is constant.

Application Cookbook

1. Use the Excel function to calculate the individual term binomial distribution probability.

2. Alternatively, use the tables printed at the end of most statistics books to find the probability of r successes in n intervals when each success has a probability p. For example, the probability of having up to two successes in three trials when each trial has a 0.2 probability is equal to 0.9920.

Binomial Distribution Table Example

Binomial TableBinomial Distribution Table

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