Code360 powered by Coding Ninjas X Naukri.com. Code360 powered by Coding Ninjas X Naukri.com
Table of contents
1.
Introduction
2.
Bernoulli Trial
3.
What is Binomial Distribution?
3.1.
Criteria for Binomial Distribution
4.
Binomial Distribution Formula
5.
Example
6.
Frequently Asked Questions
7.
Conclusion
Last Updated: Mar 27, 2024

Binomial Distribution

Master Python: Predicting weather forecasts
Speaker
Ashwin Goyal
Product Manager @

Introduction

The number of possible outcomes when we toss a fair coin, the number of possible outcomes of an examination, the number of possible outcomes if we ask people if they have been to Goa, the number of possible outcomes when we check if the sum obtained when two dice are rolled is greater than 18. The answer to all the above questions is two, also known as bi. This blog will cover Binomial Distribution, which deals with similar events. 

Bernoulli Trial

Bernoulli Trial is an experiment in which the total number of possible outcomes is exactly two. The two possible outcomes are usually success and failure, pass and fail, etc.  Each random experiment/ each Bernoulli Trial is independent of one another i.e. the probability of success is always constant over each random experiment. 

In a random experiment with n Bernoulli Trials, where all the trials are independent of each other, each trial has exactly two outcomes: success and failure and the probability of success and failure remains constant across all trials is called a Binomial Distribution.

Get the tech career you deserve, faster!
Connect with our expert counsellors to understand how to hack your way to success
User rating 4.7/5
1:1 doubt support
95% placement record
Akash Pal
Senior Software Engineer
326% Hike After Job Bootcamp
Himanshu Gusain
Programmer Analyst
32 LPA After Job Bootcamp
After Job
Bootcamp

What is Binomial Distribution?

Bi means two, Binomial Distribution is a type of probability distribution that has exactly two outcomes. For example, the tossing of a coin has exactly two possible outcomes i.e. a head or a tail. It deals with the probability of success (p) or failure (q) of an experiment or when the experiment is carried out multiple times (n = number of times the experiment is repeated).

Criteria for Binomial Distribution

Criteria for binomial distribution are as follows:

  • The number of times the experiment is carried out i.e. n is fixed.
  • Each of the observations carried out is independent of one another i.e. all the n observations are independent.
  • The probability of success = 1- The probability of failure, i.e., p = 1-q and the probability of success is constant i.e. exactly the same for all the n observations.

Once it is confirmed that the distribution follows and obeys all the criteria and that the distribution is binomial, the binomial distribution formula can be used to compute the probability.

Binomial Distribution Formula

f(x) = nCpqn-x  nCp(1-p)n-x

x = total number of successes
p = probability of success
q = probability of failure = 1-p
n = number of trials/ observations

The binomial distribution has the following properties:

  1. The mean of the distribution (μx) is equal to nP .
  2. The variance (σ2x) is nP * ( 1 - P ).
  3. The standard deviation (σx) is sqrt[ nP * ( 1 - P ) ]

Example

80% of the interns working with a company are male. If 9 interns are randomly selected, find the probability that exactly 6 are men.

Solution:
Using formula,

Here, n is the number of randomly selected interns = 9
X is the number you are asked to find the probability for = 6

Substituting the values in the first part of the formula,
n! / (n – X)!  X! = 9! / ((9 – 6)! × 6!) = 84

Now, p = probability of success = 80%
Q = probability of failure = 20%

Working the second part of the formula, p^X

= 0.8^6

= 0.262144

And, q^(n – X)

= 0.2^(9-6)

= 0.008

Putting together the values, the final answer is 0.176.

Frequently Asked Questions

  1. What is binomial distribution?
    Binomial Distribution is a type of probability distribution that has exactly two outcomes. For example, the tossing of a coin has exactly two possible outcomes i.e. a head or a tail.
  2. What are the properties of binomial distribution?
    ●   The number of times the experiment is carried out i.e. n is fixed.
    ●   Each of the observations carried out is independent of one another i.e. all the n observations are independent.
    ●   The probability of success = 1- The probability of failure, i.e., p = 1-q and the probability of success is constant i.e. exactly the same for all the n observations.
  3. Why is it called binomial distribution?
    Swiss mathematician Jakob Bernoulli, in a proof published posthumously in 1713, determined that the probability of k such outcomes in n repetitions is equal to the kth term (where k starts with 0) in the expansion of the binomial expression (p + q)n, where q = 1 − p. Hence the name binomial distribution.

Conclusion

In this article, we have extensively discussed Bernoulli trial and Binomial Distribution. We hope that this blog has helped you enhance your knowledge and if you wish to learn more, check out our Coding Ninjas Blog site and visit our Library. Do upvote our blog to help other ninjas grow.

Happy Learning!

Previous article
Uniform Distribution
Next article
Normal distribution
Live masterclass