# NCERT Solutions Class 12 Maths Chapter 13 Probability

## NCERT Solutions for Class 12 Maths Chapter 13 - Probability Refer to Probability Class 12 NCERT Solutions available on Vedantu for detailed understanding on probability, which is an important topic in Class 12 Mathematics. Students will learn the basic principles of probability and get a strong base of Chapter 13 Class 12 Maths when they refer to our solutions. Our well-designed NCERT Solutions for Class 12 Maths Chapter 13 have been prepared by knowledgeable teachers and the subject matter experts who have tried to explain this vast concept in as simple terms as possible.

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### 13.1 Introduction

In the earlier Class 12 Maths probability chapters, you have already learnt about what probability is. In any random event, the measure of uncertainty is measured by probability. A. N. Kolmogorov had formulated the axiomatic approach which was studied in the chapters. He treated probability as a function of the outcome of any experiment. In the case where there are equally likely outcomes, the relation between classical and the axiomatic theory has also been discussed. This lets you obtain the probability in the case of samples that have a discrete sample size. The addition rule in probability has also been studied.

Probability Class 12th chapter will explain the concepts of the conditional probability of any event given that the other event has occurred. This helps in understating the Bayes’ theorem, independence of events and the multiplication rule in probability. The probability distribution of random variables and the mean and the variance of probability distribution will also be studied. The binomial distribution is referred to as a discrete probability distribution which will be explained in this chapter. There are activities in the chapter that help to make the concepts clearer. The experiments are conducted assuming that there are outcomes that are equally likely.

### 13.2 Conditional Probability

In this Class 12 Probability Solutions, you learn how if there are two events that come from the same sample space, the occurrence of one of the events affects the occurrence of the other? The conditional probability of event B is the probability that the event will take place given that you already have knowledge that event A has already taken place. The probability notation is given by P(B|A) which means the probability of B given A.

In this case where the two events A and B are independent where the event A does not affect the probability of event B then the conditional probability of the event B given the event A is P(B).

However, if the two events A and B are not independent, the probability of the intersection of A and B that is the probability of both the events occurring is denoted by

P(A and B) = P(A)P(B|A).

This can help you to get the probability of P(B|A) which is obtained by

P(B|A) = P(A∩B)/P(A)

The expression is valid when P(A) is greater than 0.

Thus if A and B are two events in a sample space say S, the conditional probability of

P(A|B) = P(A∩B)/P(B)

Where P(B)>0

### 13.2.1 Properties of Conditional Probability

Property 1:If E and F are the events of a sample space say S, P(S|F) = P(F|F) = 1

Property 2: If A and B are two events in a sample space S and F is an event of S such that P(F) ≠ 0, P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F).

Property 3: P(A′|B) = 1 − P(A|B)

### 13.3 Multiplication theorem on probability

This Probability Class 12 section explains the multiplication theorem of probability. You already know that the conditional probability of event A given that B has occurred is denoted by P(A|B) and is given by:

P(A|B) = P(A∩B)P(B)

Where P(B) is not equal to 0

Thus, P(A∩B) = P(B)×P(A|B)

P(B|A) = P(B∩A)P(A)

Where P(A)  is not equal to 0.

Thus, P(B∩A) = P(A)×P(B|A)

Now Since, P(A∩B) = P(B∩A)

P(A∩B) = P(A)×P(B|A)

P(A∩B) = P(B)×P(A|B) = P(A)×P(B|A) where,

P(A)  and P(B)  is not equal to 0

The above result is known as the multiplication rule of probability.

For independent events A and B, P(B|A) = P(B). The equation then can be modified into,

P(A∩B) = P(B) × P(A)

### 13.4 Independent Events

The NCERT Solutions Class 12 Probability also explains independent events which are those events that when occurs does not affect any other event. Like if a coin is flipped in the air and the outcome is ahead. If you flip the coin again, the outcome is a tail. In both cases, the occurrence of each event is independent of the other.

The set of outcomes is known as events and this is explained well in the Probability Chapter Class 12 PDF. If the probability of an outcome of an event say A is not affected by the probability of occurrence of another event B, it is said that A and B are independent events.

### 13.5 Bayes’ Theorem

In the NCERT Solutions for Class 12 Maths Chapter 13 Probability PDF section, you learn about what Bayes’ theorem is. Bayes theorem describes the probability of an event occurring that is related to any condition. This is also considered in the case of conditional probability. Like for example, suppose you have a bag that contains three balls of different colours, say black, blue and red. You need to calculate the probability of taking out a blue ball from the bag out of the three balls. Here, the probability of the event occurring is calculated depending on the other conditions which are known as conditional probability.

### 13.5.1 Partition of a sample space – H3

This Probability NCERT Solutions Class 12 section explains what partition of sample space is. Set of events say E1, E2, …, En represents the partition of sample space S is if

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

### 13.5.2 Theorem of Total Probability

Suppose there are events, say C1, C2 . . . Cn and they form partitions of the sample space S, where all the events have a non-zero probability of occurrence. Then for any event, A associated with S, according to the total probability theorem,

P(A) = ∑k=0nP(Ck)P(A|Ck)

### Random Variables

A random variable is a variable that is capable of changing its value. The value will differ based on the varied outcomes of any experiment. If the value of any variable is dependent on the outcome from a random experiment, it is called a random variable. The random variable can take any real value.

A random variable is a real-valued function. Its domain is a sample space S of a random experiment. A random variable is capable of assigning a numerical value to every outcome in a sample space. The random variable can either be discrete or continuous. A discrete random variable is the one that assumes only specific values in an interval. Like 1.2,3… Otherwise, the random variable is a continuous random variable.

### 13.6.1 Probability Distribution of a Random Variable

The probability distribution of a random value is what defines the probability of its value that is unknown. The random variable could either be discrete or continuous. They can also be both.

There can be two random variables with an equal probability distribution and yet they can vary with how they are in respect to the other random variables or whether they are independent of the random variables. The recognition of the outcome of the value that is randomly chosen as per the probability distribution of the variable is the random variate.

For any event of any random experiment, it is possible to find its corresponding probability. Even for the different random variable values, it is possible to find its respective probability. The probability distribution of the random variable is the value of the random variable along with its corresponding probabilities.

Properties of Probability Distribution

The probability distribution of a random variable is explained in Class 12 Maths Ch 13 NCERT solutions where X is P(X = xi) = pi for x = xi and P(X = xi) = 0 for x ≠ xi.

The range of probability distribution for all possible values of a random variable is from 0 to 1, i.e., 0 ≤ p(x) ≤ 1.

### 13.6.2 Mean of a random variable

If X is a random variable and has the possible valued like x1,x2,x3,…,xn occurring with probabilities p1,p2,p3,…,pn, respectively, the mean of the random variable denoted by μ is the weighted average of the possible values of X. Each of the values is weighted by the probability with which it occurs.

The mean of a random variable can also be said as the expectation of X.

E(X) = μ = ∑i=1n xipi

=x1p1 + x2p2 + ⋯ + xnpn

### 13.6.3 Variance of a Random Variable

The variance tells about how spread out the X values is around its mean. The variance of a random variable is denoted by σ2x with the values of X as x1,x2,x3,…,xn and their occurrence with the probability of p1,p2,p3,…,pn.

Var(X) = σ2x = ∑i=1n(xi − μ)2pi

Var(X) = ∑i=1n (xi)2pi + ∑i=1n μ2pi − ∑o=1n 2xiμpi

Var(X) = ∑i=1n (xi)2pi + μ2 ∑ ni=1 pi − 2μ∑i=1n xipi

Here,

∑i=1nxipi = μ (Mean of X) and ∑i=1n pi = 1 (sum of probabilities of all the outcomes of an event is 1). Substituting the values, we get

Var(X) = ∑i=1n(xi)2pi + μ2 − 2μ2

σ2x = Var(X) = ∑i=1n (xi)2pi − μ2

Var(X) = E(X2) – [E(X)]2

Where,

E(X2) = ∑i=1n(xi)2pi and E(X) = ∑i=1nxipi

### 13.7 Bernoulli Trials and Binomial Distribution

This Ch 13 Maths Class 12 section explains Bernoulli trials and the binomial distribution in brief. Bernoulli trial or binomial trial is where there are only two possible outcomes. In the case of the binomial distribution, we can get the number of successes in the sequence of experiments that are independently conducted.

### 13.7.1 Bernoulli Trials

There are many random experiments that can have only two of the possible outcomes. These are success and failure. Like for example, the toss of a coin which can be heads or tails or the result that something can either be defective or not defective.

These are independent trials and can have only two of the possible outcomes. These are known as the Bernoulli trials. Here are the conditions of a trial to be categorized into a Bernoulli trial.

• The number of trials should be finite

• The trials should be independent

• Each trial should have only two outcomes with are success and failure

• The probability of either success or failure will not change for every trail

### 13.7.2 Binomial Distribution

Suppose there is a random experiment that can exactly have only two outcomes and this is repeated for n number of times and independently. The probability of success is denoted by p and the probability of failure is denoted by q.

Out of these trials, we get success in x number of trials and we get failure in the remaining (n-x) number of trials. Thus the number of trials in which we can have success is nCx

A random variable has a binomial distribution when

P(X = x) = p(x) = nCx px qn-x,

for x = 0, 1, … , n and P(X = x) = 0 otherwise. Here, q = 1 – p. The random variable X is a binomial variable. Thus you can say that a binomial trial is a set of Bernoulli trials which has n independent trials.

### Here are the Conditions for a Binomial Distribution.

• Each trial should result in only two outcomes which are success and failure

• The number of trials says n is finite

• Each trial is independent of each other

• The probability of success which is p and the probability of failure which is q stay constant in each trial.

### Key Features of the NCERT Solution for Class 12 Maths Chapter 13

Probability is an important topic which will be studied further if you take up mathematics at the undergraduate level. This is why it is important to build on your concepts and clear your doubts right from the start. This is why there are Vedantu’s expert teachers who have solved the NCERT probability Class 12 questions with a complete explanation to ensure that you do not just learn the concept for your examination but also master the topic.

• The Ch 13 Class 12 Maths  NCERT solution by Vedantu prepares you well for your examination.

• The solutions are clear and detailed to give you a thorough explanation of the topic

• It is important to keep referring to these solutions to understand the various approaches to the questions.

• Solving these questions and checking out the solutions will make you more confident in attempting the probability question in the examination.

1. How many topics and sub-topics are there in this chapter?

A. Studying Maths chapter 13 of Class 12 namely Probability helps the students to understand various relevant concepts such as Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution etc.

Following concepts are covered in Chapter 13 of NCERT Solutions for Class 12. Take a look at the following list:

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

2. What is Conditional probability?

A. Conditional probability basically depicts the probability of an event taking place with some relationship to one or more events. Suppose E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E will be under the condition of which the event F has occurred, written as P (E | F).

3. How many exercises are there in Class 12 Maths Chapter 13?

A. Chapter 13 Probability NCERT Solutions contains several exercises. Every exercise contains the answer to each question that has been asked in every exercise of chapter 13. All the answers are accurate and provided with step-by-step instructions. This is considered as the most helpful study material for every student to do their home assignments as well as practice various sample papers. Here’s the number of questions of all the exercises -

• Chapter 13 Exercises 13.1 - 17 questions

• Chapter 13 Exercises 13.2 - 18 questions

• Chapter 13 Exercises 13.3 - 14 questions

• Chapter 13 Exercises 13.4 - 17 questions

• Chapter 13 Exercises 13.5 - 15 questions

4. Why Vedantu’s NCERT Solutions for Class 12 Maths Chapter 13 is beneficial for the students?

A. NCERT Solutions for Class 12 Maths Chapter 13 Probability is provided in PDF format on our site which you can download very easily as per your convenience. These solutions are designed and prepared by the best teachers and subject matter experts from the respective fields. All the important topics are covered in every answer to the questions from the exercises of the NCERT textbook and each answer comes with a detailed step-by-step explanation to help students understand concepts better. These NCERT solutions play a crucial role in building the foundation stronger and also assist in your preparation for other competitive exams such as JEE Main, JEE Advanced, Olympiad.

Chapter 13 Probability of the NCERT Solutions for Class 12 Maths covers multiple exercises which are given in between and at the end of the chapter. Check our website or download our mobile app to access the NCERT Solutions in PDF downloadable format. SHARE TWEET SHARE SUBSCRIBE