Binomial Distribution

Bernoulli Trial

A single experiment or observation with only two possible outcomes; success or failure.

Binomial distribution

The probability distribution of the number of successes in a fixed number of independent Bernoulli trials,

each with the same success probability p.

The experiment consists of N independent Bernoulli trials

First requirement for a Binomial Distribution

Each trial has the same probability p of success

Second requirement for a Binomial Distribution

The random variable X counts the number of successes in n trials.

Third requirement for a Binomial Distribution

Notion for Binomial Distribution

The random variable X follows a binomial distribution if X ~ Bin(n, p). Possible values: 0, 1, 2, …, n.

independent

Trials are ____________ if the outcome of one doesn’t affect the outcomes of others.

finite population

When sampling without replacement from a _________________, the trials are not strictly independent because each draw affects the next.

Approximate independence condition

If the population is very large compared to the sample (so the sample is small), the draws are nearly independent, and the binomial model can be used as an approximation.

Rule of Thumb (5% condition)

When the sample size is no more than 5% of the population, the binomial distribution can be used to model the number of successes, even though sampling is without replacement.

X

Symbol that represents the number of successes in n Bernoulli trials. It is the random variable in Binomial context.

Example of a Binomial setup

Sampling several components from a large lot and counting how many are defective; each test whether it is defective or not is a Bernoulli trial, and the large number of defective follows a binomial distribution