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# NCERT Solutions for Class 12 Maths Chapter 13 Free PDF Download

## NCERT Solutions for Class 12 Maths Chapter 13 – Probability

NCERT Solutions for Class 12 Maths Chapter 13 will provide complete knowledge about the probability. Our experts of the subject have done hard work to give solutions to the numerical of this chapter in the easiest way. Their efforts are giving the result in the form of NCERT solutions.

NCERT solutions for class 12 Maths for Chapter 13 Probability are very helpful in providing the strong concept of finding the possibility of occurrence of an event. NCERT solutions will give you a strong base of skill for finding solutions to questions of probability. These solutions will also help you with your homework and best performance in exams.

Our team of dedicated teachers has created NCERT solutions for class 12 Maths for Chapter 13 Probability according to the syllabus of CBSE. Toppr app will help you to get complete NCERT solutions.

### CBSE Class 12 Maths Chapter 13 – Probability NCERT Solutions

NCERT solutions for class 12 Maths for Chapter 13 Probability will help the students to understand the purpose of probability and its applications in real life. It presents the solutions in a very effective easy way. In this chapter, the student will learn the different category of problems and their required solutions. The conceptual background of probability is very much necessary for other branches of mathematics as well as science. ### Sub-topics covered under NCERT Solutions for Class 12 Maths Chapter 13

• 13.1 Introduction
• 13.2 Conditional Probability
• 13.2.1 Properties of conditional probability
• 13.3 Multiplication Theorem on Probability
• 13.4 Independent Events
• 13.5 Bayes’ Theorem
• 13.5.1 Partition of a sample space
• 13.5.2 Theorem of total probability
• 13.6 Random Variables and its Probability Distributions
• 13.6.1 Probability distribution of a random variable
• 13.6.2 Mean of a random variable
• 13.6.3 Variance of a random variable
• 13.7 Bernoulli Trials and Binomial Distribution
• 13.7.1 Bernoulli trials
• 13.7.2 Binomial distribution

### NCERT Solutions for Class 12 Maths Chapter 13

Probability is very important in terms of statistical problems based on chance. This chapter covers all fundamental aspects with various theorems and formula. NCERT solutions for class 12 Maths for Chapter 13 Probability contains many examples and problems to give maximum practice to the students. Many tricks will help students to get the solution easily.

Let us discuss the sub-topics in detail.

13.1 Introduction

The probability is as a measure of uncertainty of events in a random experiment. We shall discuss the important concept of the conditional probability of an event if another event has already occurred. This will be helpful in understanding the Bayes’ theorem and multiplication rule of probability.

13.2 Conditional Probability

This section will explain that if we have two events from the same sample space, then what be the effect on the probability of occurrence of one of the events due to the occurrence of the other event.

13.2.1 Properties of conditional probability

This section elaborates various properties applicable to finding the conditional probability. Any relevant examples will help the student to get the real concept.

13.3 Multiplication Theorem on Probability

This topic will explain the procedure to find the probability of the occurrence of two events together. This is based on multiplication theorem of probability. Also, the student will apply it for more than two events also.

13.4 Independent Events

Independent events are the events such that the probability of occurrence of one of them does not affect the occurrence of the other. For example, rolling a dice two times.

13.5 Bayes’ Theorem

John Bayes’ solved the problem of finding the reverse probability by using conditional probability. It is a very popular theorem used in probability.

13.5.1 Partition of a sample space

The events E1, E2, …, En represent a partition of the sample space S if these are pairwise disjoint, exhaustive and have a nonzero probability.

13.5.2 Theorem of total probability

This theorem helps students to find the total probability for a sample space with many events of equal partition.

13.6 Random Variables and its Probability Distribution

As we know that a random variable is a real-valued function whose domain is the sample space obtained from a random experiment. Its probability will be based on probability distributions.

13.6.1 Probability distribution of a random variable

This section elaborates the description giving the values of the random variable along with the corresponding probabilities. This is the probability distribution of a random variable X.

13.6.2 Mean of a random variable

Mean is a measure of location or central tendency in statistics. It will roughly locate a middle or average value of the random variable.

13.6.3 Variance of a random variable

Variance represents the deviation of values from their mean. Also for a small value of variance, the value of the random variable is close to the mean.

13.7 Bernoulli Trials and Binomial Distribution

This section discusses some experiments which are dichotomous in nature and elaborates the Bernoulli principle.

13.7.1 Bernoulli trials

The student will come to know about independent trials which have only two outcomes usually referred to as ‘success’ or ‘failure’. These are the Bernoulli trials.

13.7.2 Binomial distribution

Binomial expansion is used to find the probability distribution of a number of successes in an experiment consisting of n Bernoulli trials.

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NCERT solutions for class 12 Maths for Chapter 13 Probability is very popular among the students because it helps them for finding the solution of complex problems in maths and statistics both. Our solution will provide the best presentation of the possible solutions and their methods.

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