NCERT Class 12-Mathematics: Chapter – 13 Probability Part 1 (For CBSE, ICSE, IAS, NET, NRA 2023)

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13.1 Overview

13.1. 1 Conditional Probability

If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the condition that the event F has occurred, written as , is given by

13.1. 2 Properties of Conditional Probability

Let E and F be events associated with the sample space S of an experiment. Then:


(ii) , where A and B are any two events associated with S.


13.1. 3 Multiplication Theorem on Probability

Let E and F be two events associated with a sample space of an experiment. Then

If and G are three events associated with a sample space, then

13.1. 4 Independent Events

Let E and F be two events associated with a sample space S. If the probability of occurrence of one of them is not affected by the occurrence of the other, then we say that the two events are independent. Thus, two events E and F will be independent, if



Using the multiplication theorem on probability, we have


Three events A, B and C are said to be mutually independent if all the following conditions hold:


13.1. 5 Partition of a Sample Space

A set of events is said to represent a partition of a sample space S if


(b) , and

(c) Each

13.1. 6 Theorem of Total Probability

Let be a partition of the sample space S. Let A be any event associated with S, then

13.1. 7 Bayes՚ Theorem

If are mutually exclusive and exhaustive events associated with a sample space, and A is any event of non-zero probability, then

13.1. 8 Random Variable and its Probability Distribution

A random variable is a real valued function whose domain is the sample space of a random experiment.

The probability distribution of a random variable X is the system of numbers

Random Variable and Its Probability Distribution


13.1. 9 Mean and Variance of a Random Variable

Let be a random variable assuming values with probabilities respectively such that . Mean of , denoted by [expected value of X denoted by ] is defined as

and variance, denoted by , is defined as

Standard deviation of the random variable is defined as

13.1. 10 Bernoulli Trials

Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:

(i) There should be a finite number of trials

(ii) The trials should be independent

(iii) Each trial has exactly two outcomes: success or failure

(iv) The probability of success (or failure) remains the same in each trial.

13.1. 11 Binomial Distribution

A random variable taking values is said to have a binomial distribution with parameters and , if its probability distribution is given by