Binomial Distribution
In statistics and probability theory, the binomial distribution is the probability distribution that is discrete and applicable to events having only two possible results in an experiment, either success or failure. (the prefix “bi” means two, or twice). To give an example, if we toss a coin, there are only two possible outcomes: heads or tails. Similarly, if any test is taken, then there are distinct two results: pass or fail. Such distribution is also called a binomial probability distribution.
Two parameters n and p are used here in a binomial distribution. The variable ‘n’ represents the number of times the experiment runs and the variable ‘p’ states the probability of any one(each) outcome. Let us consider an example to understand this better. If a die is thrown randomly 10 times, then the probability of getting a 3 for any throw is 1/6. Similarly, if we will throw the dice 10 times, we will have n = 10 and p = 1/6. In this article let us learn the formula to calculate the two outcome distribution (Binomial distribution) among multiple experiments with solved examples.
What Is Binomial Distribution?
The binomial distribution is a commonly used discrete distribution used in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment.
Binomial Distribution in Statistics
The binomial distribution forms the base for the famous binomial test of statistical importance. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the booleanvalued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution.
Negative Binomial Distribution
Let's understand with an example about when can a binomial distribution be negative. Suppose we throw a dice and determine that the occurrence of 2 will be a failure and all non2’s will be successes. Let the failures be denoted by ‘r’. Now, if the dice is thrown frequently until 2 appears the third time, i.e., r = three failures, then the binomial distribution of the number of non2's that arrived would be the negative binomial distribution.
Binomial Distribution Examples
We now already know that binomial distribution gives the probability of a different set of outcomes. In real life, the concept of the binomial distribution is used for:
 Finding the quantity of raw and used materials while making a product.
 Taking a survey of positive and negative reviews from the public for any specific product or place.
 By using the YES/ NO survey
 To find the number of male and female students in a university.
 The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.
Binomial Distribution Formula
The binomial distribution formula is for any random variable X, given by; P(x:n,p) = ^{n}C_{x} p^{x }(1p)nx Or P(x:n,p) = ^{n}C_{x} p^{x} (q)nx
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 in the form of nBernoulli trials, where ^{n}C_{x} = n!/x!(nx)!. Hence, P(x:n,p) = n!/[x!(nx)!].p^{x}.(q)^{nx}
Binomial Distribution Mean and Variance
For a binomial distribution, the mean, variance, and standard deviation for the given number of success are represented using the formulas
 Mean, μ = np
 Variance, σ^{2 }= npq
 Standard Deviation σ= √(npq)
Where p is the probability of success q is the probability of failure, where q = 1p
Binomial Distribution Vs Normal Distribution
The main difference between the binomial distribution and the normal distribution is that the binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.
Properties of Binomial Distribution
The properties of the binomial distribution are:
 There are only two distinct possible outcomes: true/false, success/failure, yes/no.
 There is a fixed number of 'n' times repeated trials in a given experiment.
 The probability of success or failure varies for each attempt/trial.
 Only the successful attempts are calculated out of 'n' independent trials.
 Every trial is an independent trial on its own, this means that the outcome of one trial has no effect on the outcome of another trial.
Related Topics
Important Notes
 For using the binomial distribution, the number of observations or trials in an experiment is fixed.
 Each observation/attempt/trial is independent on its own. This means none of the trials have an effect on the probability of the next trial.
 Each trial has an equal probability of occurrence. The probability of success is exactly the same from one trial to another.
Solved Examples on Binomial Distribution

Example 1: If a coin is tossed 5 times, using binomial distribution find the probability of:
(a) Exactly 2 heads
(b) At least 4 heads.
Solution:
(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:
Number of trials: n=5
Probability of head: p= 1/2 and hence the probability of tail, q =1/2
For exactly two heads:
x=2
P(x=2) = ^{5}C2 p^{2} q^{52 }= 5! / 2! 3! × (½)^{2}× (½)^{3}
P(x=2) = 5/16
(b) For at least four heads,
x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)
Hence,
P(x = 4) = ^{5}C4 p^{4} q^{54} = 5!/4! 1! × (½)^{4}× (½)^{1} = 5/32
P(x = 5) = ^{5}C5 p^{5} q^{55} = (½)^{5} = 1/32
Answer:Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, find the probability of getting at most 2 heads.
Solution:
Solution: P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)
P(X = 0) = (½)5 = 1/32
P(X=1) = 5C1 (½)5.= 5/32
Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16
FAQs on Binomial Distribution
What Is a Binomial Distribution in Statistics?
The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial.
What Is the Purpose of Binomial Distribution?
The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.
What Is the Formula for Binomial Distribution?
The formula for binomial distribution is:
P(x: n,p) = _{n}C^{x} p^{x} (q)^{nx}
Where p is the probability of success, q is the probability of failure, n = number of trials.
What Is the Formula for the Mean and Variance of the Binomial Distribution?
The mean and variance of the binomial distribution are:
Mean = np
Variance = npq
Where p is the probability of success, q is the probability of failure, n = number of trials.
What Are the Criteria for the Binomial Distribution?
The criteria for using the binomial distribution are:
 The number of trials should be fixed.
 Each trial should be independent.
 The probability of success is exactly the same from one trial to the other trial.
What Is the Difference Between a Binomial Distribution and Normal Distribution?
The main difference between the binomial distribution and the normal distribution is that the binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.
How Do you Identify a Binomial?
For a variable to be a binomial random variable, all of the following conditions must be met:
 There are a fixed number of trials (a fixed sample size).
 On each trial, the event of interest either occurs or does not.
 The probability of occurrence (or not) is the same on each trial.
 Trials are independent of one another.