# Binomial Distribution Formula

## Binomial Distribution Formula

In statistics, the binomial distribution is a discrete distribution that is frequently employed. In contrast to a binomial distribution, the normal distribution is continuous. Given a success probability of “p” for each trial during the experiment, the binomial distribution represents the likelihood that an experiment will yield “x” successes out of “n” trials.

The binomial distribution is the foundation of the well-known binomial test of statistical significance. A Bernoulli process is a test with a string of results, and a Bernoulli experiment is a test with a single outcome, such as success or failure. Imagine an experiment where a yes-or-no response is required for each of n questions. The success/yes/true/one result is therefore represented with probability p in the binomial probability distribution, and the failure/no/false/zero outcome with probability q (q = 1 p). When n = 1, the binomial distribution is known as a Bernoulli distribution in a single experiment.

## What Is the Binomial Distribution Formula?

The binomial distribution formula is for any random variable X, given by; P(x:n,p) = nCxx px (1-p)n-x Or P(x:n,p) = nCx px (q)n-x

where,

• n = the number of experiments
• x = 0, 1, 2, 3, 4, …
• p = Probability of success in a single experiment
• q = Probability of failure in a single experiment (= 1 – p)

The binomial distribution formula is also written as n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

## Solved Examples on Binomial Distribution Formula

Example 1:

A coin is tossed 12 times. What is the probability of getting exactly 7 heads?

Solution:

Given that a coin is tossed 12 times. (i.e.) n= 12

Thus, the probability pf getting head in a single toss = ½. (i.e.) p = ½.

So, 1-p = 1-½ = ½.

The binomial probability distribution is P(r) = NCR · pr (1 − p)n−r.

Now, we have to find the probability of getting exactly 7 heads. (i.e.) r = 7.

Substituting the values in the binomial distribution formula, we get

P(7) = 12C7 · (½)7 (½)12−7

P(7) = 792· (½)7 (½)5

P(7) = 792.(½)12

P(7) = 792 (1/4096)

P(7) = 0.193

Therefore, the probability of getting exactly 7 heads is 0.193.