Binomial Distribution

In probability theory and statistics, the binomial distribution is the discrete probability distribution which gives only two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.

There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙.

Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.

Also, read:

Binomial Probability Distribution

In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the boolean-valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution. The binomial distribution is the base for the famous binomial test of statistical importance.

Negative Binomial Distribution

In probability theory and statistics, the number of successes in a series of independent and identically distributed Bernoulli trials before a particularised number of failures happens. It is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e., r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.

Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In real life, the concept 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, we can check whether the number of persons views the particular channel.
  • To find the number of male and female employees in an organisation.
  • 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) = nCx px (1-p)n-x


P(x:n,p) = nCx px (q)n-x


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 can also be written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence,

P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

Properties of Binomial Distribution

The properties of the binomial distribution are:

  • There are two possible outcomes: true or false, success or failure, yes or no.
  • There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
  • The probability of success or failure varies for each trial.
  • Only the number of success is calculated out of n independent trials.
  • Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.

Binomial Distribution Examples And Solutions

Example 1: If a coin is tossed 5 times, find the probability of:

(a) Exactly 2 heads

(b) At least 4 heads.


(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:


P(x=2) = 5C2 p2 q5-2 = 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)


P(x = 4) = 5C4 p4 q5-4 = 5!/4! 1! × (½)4× (½)1 = 5/32

P(x = 5) = 5C5 p5 q5-5 = (½)5 = 1/32


P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, find the probability of:

a) Getting at least 2 heads

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


P(X ≤ 2) = 1/32 + 5/32 = 3/16

Probability is a wide and very important topic for class 11 and class 12 students. By capturing the concepts here at BYJU’S, students can excel in the exams.

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