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Independent & Mutually Exclusive Events in Probability-Definition, Venn Diagram & Solved Examples | Class 12 Math Notes Study Material Download Free PDF

Independent events are those events whose occurrence is not dependent on any other event. For example, if we flip a coin in the air and get the outcome as Head, then again if we flip the coin but this time we get the outcome as Tail. In both cases, the occurrence of both events is independent of each other. It is one of the types of events in probability. Let us learn here the complete definition of independent events along with its Venn diagram, examples and how it is different from mutually exclusive events.

Class 12 Math Chapter Probability: Important Topics for CBSE Board Exam Students

Written by Author Neeraj Anand (Director of Anand Classes Jalandhar)
Published by Anand Technical Publishers

What are Independent Events?

In Probability, the set of outcomes of an experiment is called events. There are different types of events such as independent events, dependent events, mutually exclusive events, and so on.

If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events.

Consider an example of rolling a die. If A is the event ‘the number appearing is odd’ and B be the event ‘the number appearing is a multiple of 3’, then

P(A)= 3/6 = 1/2 and P(B) = 2/6 = 1/3

Also A and B is the event ‘the number appearing is odd and a multiple of 3’ so that

P(A ∩ B) = 1/6

P(A│B) = P(A ∩ B)/ P(B)

=1613=12

P(A) = P(A│B) = 1/2  , which implies that the occurrence of event B has not affected the probability of occurrence of the event A .

If A and B are independent events, then P(A│B) = P(A)

Using the Multiplication rule of probability, P(A ∩ B) = P(B) .P(A│B)

P(A ∩ B) = P(B) .P(A)

Note: A and B are two events associated with the same random experiment, then A and B are known as independent events if P(A ∩ B) = P(B) .P(A)

What are Mutually Exclusive Events?

Two events A and B are said to be mutually exclusive events if they cannot occur at the same time. Mutually exclusive events never have an outcome in common.

Independent Events Vs Mutually Exclusive Events

The difference between the independent events and mutually exclusive events are given below:

Independent EventsMutually exclusive events
They cannot be specified based on the outcome of a maiden trial.They are independent of trials
Can have common outcomesCan never have common outcomes
If A and B are two independent events, then P(A ∩ B) = P(B) .P(A)If A and B are two mutually exclusive events, then P(A ∩ B) = 0

Independent Events Venn Diagram

Let us proof the condition of independent events using a Venn diagram.

Theorem: If X and Y are independent events, then the events X and Y’ are also independent.

Proof: The events A and B are independent, so, P(X ∩ Y) = P(X) P(Y).

Let us draw a Venn diagram for this condition:

Independent & Mutually Exclusive Events in Probability
From the Venn diagram, we see that the events X ∩ Y and X ∩ Y’ are mutually exclusive, and together they form the event X.

X = ( X ∩ Y) ∪ (X ∩ Y’)
Also, P(X) = P[(X ∩ Y) ∪ (X ∩ Y’)]
or, P(X) = P(X ∩ Y) + P(X ∩ Y’)
or, P(X) = P(X) P(Y) + P(X ∩ Y’)
or, P(X inter Y’) = P(X) − P(X) P(Y) = P(X) (1 – P(Y)) = P(X) P(Y’)

Example with Solution

Question: Let X and Y are two independent events such that P(X) = 0.3 and P(Y) = 0.7. Find P(X and Y), P(X or Y), P(Y not X), and P(neither X nor Y).

Solution: Given P(X) = 0.3 and P(Y) = 0.7 and events X and Y are independent of each other.

P(X and Y) = P( X ∩ Y) = P(X) P(Y) = 0.3 × 0.7 = 0.21

P(X or Y) = P(X ∪ Y) = P(X) + P(Y) – P(X ∩ Y) = 0.3 + 0.7 – 0.21 = 0.79

P(Y not X) = P(Y ∩ X’) = P(Y) – P(X ∩ Y) = 0.7 – 0.21 = 0.49

And P(neither X nor Y) = P(X’ ∩ Y’) = 1 – P(X ∪ Y) = 1 – 0.79 = 0.21


For CBSE Class 12 students, the chapter on Probability is not just an important part of the curriculum but also a fascinating one. It is essential for those who want to understand the concepts of chance and uncertainty and apply them in real-life situations, as well as in competitive exams like JEE. To help you conquer this chapter, Neeraj Anand, Director of Anand Classes, Jalandhar, has written a detailed guide that will help students excel in Probability. Published by Anand Technical Publishers, this book simplifies complex concepts and prepares students for success in their CBSE Board Exams.

Key Topics in Class 12 Math Probability for CBSE:

  1. Introduction to Probability: Understanding the basic concepts of probability, including sample space, events, and outcomes.
  2. Conditional Probability: Learn how to calculate the probability of an event given that another event has already occurred.
  3. Multiplication Theorem of Probability: Master the multiplication rule to find the probability of the intersection of events.
  4. Bayes' Theorem: Understand and apply Bayes' Theorem to solve conditional probability problems.
  5. Random Variables and Probability Distributions: Learn about random variables, their types, and probability distributions (binomial, Poisson, etc.).
  6. Mean and Variance of Random Variables: Study the mean and variance of random variables to understand the expected value and spread of data.
  7. Binomial Distribution: Explore the properties and applications of binomial distribution in probability theory.
  8. Probability Density Function: Learn how to work with continuous random variables using probability density functions.

Why This Book is Essential:

  • Clear Explanation of Concepts: Neeraj Anand has an exceptional ability to break down complex probability concepts into easy-to-understand steps, ensuring that students grasp every concept thoroughly.
  • Numerous Practice Problems: The book includes a variety of practice questions with detailed solutions to help reinforce understanding and build problem-solving skills.
  • CBSE Focused Content: Aligned perfectly with the CBSE syllabus, the book helps students prepare for the exact questions they will face in their board exams.
  • Student-Centered Approach: With real-world examples and simple explanations, the book makes the study of probability both engaging and applicable to daily life.

Perfect for CBSE Board Exam Preparation:

Probability can seem daunting at first, but with the right guidance, it becomes a manageable and interesting topic. Neeraj Anand’s book provides students with all the tools and techniques they need to confidently tackle this chapter and excel in the CBSE Board Exams.

Grab Your Copy Today!

Get your copy of Class 12 Math Chapter Probability by Neeraj Anand and take the first step towards mastering this essential chapter. Available at all leading bookstores or through Anand Technical Publishers.

With Neeraj Anand's expert guidance, you’ll be able to ace Probability and build a solid foundation for future studies in mathematics, statistics, and even fields like economics and data science.

Make Probability your strength for the CBSE Board Exams! 🌟

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