6.7: Binomial Probability Distribution

A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.

The outcomes of a binomial experiment fit a binomial probability distribution. A statistical experiment can be classified as a binomial experiment if the following conditions are met:

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.

There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of success on one trial, and q denotes the probability of failure on one trial. p + q = 1.

The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of success and probability, q, of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If success is guessing correctly, then failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success (p = 0.6) and his probability of failure (q = 0.4) remain the same.

This text is adapted from Openstax, Introductory Statistics, Section 4.3, Binomial Distribution

Binomial probability distribution represents cases that have multiple but fixed number of trials, like in a coin-toss, with two possible outcomes per trial.

Here n denotes the number of trials.

In each trial, the probability of success, heads, is denoted by p, whereas the probability of failure, tails, is represented by q. If one is known, the other can be easily calculated.

For a binomial distribution, the probability of success or failure should always be the same for all the trials.

Also, the outcome of each trial must be independent of other trials.  

In this example, the number of heads is the random variable, x, whose value can be a whole number between 0 and n.

P of x denotes the probability of x heads among n trials, calculated using the Binomial probability formula.

Here, the factorial symbol represents the product of decreasing factors.

For each value of x, P of x can be obtained, which can be plotted to get the graphical form of the binomial distribution.