CBSE Class 12 Maths Revision Notes Chapter 13

Class 12 Mathematics Chapter 13 Notes

Mathematics plays an essential role in the success of students appearing for the Class 12 boards and other national level competitive exams. To excel in the subject, students must practice regularly and plan their studies very well. Class 12 Mathematics is further divided into six units. Students can score a maximum of 10 marks from the probability unit. The CBSE syllabus of Mathematics is precise and delivers knowledge in an  easy manner. 

The Class 12 Mathematics Chapter 13 notes reviews all basic formulas introduced in Class 11 and explain the advanced concepts of probability. These revision notes compile all solutions to the exercise questions from the chapter. Students can access these Class 12 Mathematics Chapter 13 notes to strengthen their fundamental concepts and clear the logic of each problem effectively.  

Extramarks, an online learning platform, aims to provide a productive learning experience through academic notes such as the Class 12 Mathematics Chapter 13 notes to all students.

 Class 12 Mathematics Chapter 13 Notes: Key Topics

The following topics are explained in detail in the class 12 Mathematics chapter 13 notes:

• Introduction
• Basic Terminology
• Conditional Probability
• Properties of conditional probability
• Multiplication Theorem on Probability
• Independent Events
• Bayes’ Theorem
• Random Variables and Their Probability Distribution
• Mean and Variance of a Random Variable
• Bernoulli Trials
• Binomial Distribution

Class 12 Mathematics chapter 13 notes: An Overview 

Introduction:

In this chapter, the important concept of the conditional probability of an event with respect to the occurrence of another event is explained in detail, which is helpful in understanding the concepts like Bayes’ theorem, multiplication rule and independent events in a sample space. The concept of a random variable, its probability distribution, mean and variance is very well explained in the Class 12 Mathematics Chapter 13 notes. The last section of the chapter discusses the discrete probability distribution based on Bernoulli trials, called Binomial distribution. 

What is Probability? 

Probability measures the chance is there for an expected result to occur. We can say that Probability is equal to the number of favourable outcomes divided by the total no. of outcomes.

Therefore, Probability = number of favourable outcomes / total no. of outcomes.

For example: If a basket has balls of three colours: Red, Green, and Blue, the Probability of picking a red ball will be,

Total number of possibilities = 3 (The ball we pick can be Red, Green or blue)

Total number of favourable possibilities to get a red ball = 1

Therefore, the probability of selecting a red ball is P=1/3

With the help of the Class 12 Mathematics Chapter 13 notes, students may learn the most  about probability. 

Basic Terminology

1. Experiment: 

An experiment is a function that gives well-defined results.

      2. Random Experiment:

A random experiment is carried out where the outcome may not be the same, even in identical conditions.

      3. Sample space: 

The Sample space, denoted by S, is a set of all the possible outcomes of a random experiment.

1. Sample point: 

Every element included in the sample space is known as the sample point. For example, When a coin is tossed, the sample space is given as S={HEAD, TAIL} and the elements, namely head or tail, are called the sample points.

1. Event: 

The set of all favourable outcomes is an event. It is a subspace of the sample set. For, e.g., Suppose you toss a coin and are looking for the head, then the head is the favourable outcome.

1. Mutually Exclusive Events: 

Any two events are called mutually exclusive if there is no common element present between them.

1. Equally Likely Events: 

It is also known as the relative property of two events. Consider any two events. They are termed to be equally likely if none of the events will occur in preference to the other. For example, in tossing an unbiased coin, both heads and tails are equally likely to occur. 

1. Independent Event: 

For any two events, when the occurrence or non-occurrence of one event does not depend on the occurrence or non-occurrence of another event, they are said to be independent.

1. Exhaustive events: 

Exhaustive events are a set of events in a sample space (S) such that at least one of them compulsorily occurs. 

1. Complement of an Event: 

If S is a sample space and B is an event in it, then the complement of B is represented as B’ or B where B’= {x: x ∈ S, x ∉ B}.

Refer to the Class 12 Mathematics Chapter 13 notes to understand each terminology clearly. 

Remarks: 

• If the probability of a given event is One, it does not indicate that it will surely occur. 
• Probability just predicts the occurrence of one event in contrast to other given events. 
• These predictions are based on data and the method of analysing it. 
• Furthermore, if the probability of a given event is zero, it does not indicate that it will never occur. 

Conditional Probability: 

The conditional Probability is termed as the probability that an event B will take place provided that an event A has previously occurred.

This Probability is denoted by P(B|A) and is read as the probability of B given A. If events A and B are independent, the conditional probability is given as the probability of event B, i.e., P. In this case, the probability of event B, i.e., (P(B)) is independent of event A. 

If A and B events are not independent, then the probability that both events will occur is given by 

P (A & B) = P(A) P(B|A)

Consider a random experiment having two sample spaces, E and S. These two events belong to the same sample space, then the conditional probability of the event E given that event S has occurred, P(E|S) is defined as P(E|S)=P(E∩S)P(S), provided P(S) ≠ 0.

Conditional probability is also defined as,

P(E|S)= Number of events which are favourable to E∩S Number of events which are favourable to S = n(E∩S)n(S)

The concept of conditional probability is very well explained in the Class 12 Mathematics Chapter 13 notes. 

Properties:

If E and F and events associated with sample space S, then the following properties hold: 

Property 1:

P(S|F) = 1 and P(F|F) = 1

We know that, P(S|F) = P(S∩F)P(F) = P(F)P(F)= 1

and P(F|F) = P(F∩F)P(F) = P(F)P(F)= 1

Therefore, P(S|F) = 1 and P(F|F) = 1

Property 2:

Consider two events, M and N, of a sample space S and let F be an event of the same sample space, such that P(F)≠ 0, then P(M∪N)|F) = P(M|F) + P(N|F) – P(M∩N)|F)

If M and N are disjoint then, since P(M∩N)|F) = 0, as a result, P(M∪N)|F) = P(M|F) + P(N|F). 

We know that P(M∪N)|F) = P(M∪N)∩F )P(F) 

Using distributive law, we get P(M∪N)|F) = P((M∩F) ∪ (N∩F))P(F) = P(M∩F) + P (N∩F) – P (M∩N∩F) P(F)

This can be written as, P(M∪N)|F) = P(M∩F)P(F)+ P (N∩F)P(F)–P (M∩N∩F)P(F) 

Therefore, P(M∪N)|F) = P(M|F) + P(N|F) – P(M∩N)|F)

Property 3:

P(E’|F) = 1 – P(E|F)

Using Property 1, P(S|F) = 1, we get

P(E E’ |F) = 1 as S = E E’ 

Since events E and E’ are disjoint, we can write P (E|F) + P(E’|F) = 1

Therefore, P(E’|F) = 1 – P(E|F)

Students should practice the derivations of each property given in the Class 12 Mathematics Chapter 13 notes. 

Multiplication Theorem:

Consider two events, E and F, in a sample space of an experiment, the intersection of the two events is given as P(E∩F) = P(E) P(F|E), where P(E) ≠ 0

                  = P(F) P(E|F), where P(F) ≠ 0

The conditional probability of event given event F has occurred, i.e., P(E|F) is given as

P(E|F) = P(E ∩ F)P(F), provided P(F)0. 

As a result, P(E ∩ F)= P(E|F). P(F)….. Equation 1

Now, P(F|E) = P(F ∩ E)P(E), provided P(E)0

Since F ∩ E = E ∩ F, we get

P(F|E) = P(E ∩ F)P(E), provided P(E)0

As a result, P(E ∩ F)= P(F|E). P(E)….. Equation 2

Combining equations 1 and 2, we get

P(E∩F) = P(E) P(F|E), where P(E) ≠ 0

             = P(F) P(E|F), where P(F) ≠ 0

Consider three events, E, F and, in a sample space, then the intersection of the three events is given as P(E∩F∩G)= P(E) P(F|E) P(G|E∩F)

                  = P(E) P(F|E) P(G|EF)

NOTE: 

The multiplication rule of probability is extended to more than three events as well. 

The Class 12 Mathematics Chapter 13 notes include several practice problems on the multiplication theorem. 

Independent Events:

Consider two events, M and N, in a sample space (S) of a random experiment. If the probability of occurrence of one of the events is not affected or is independent with respect to the occurrence of the other event, then we can say that the two events, M and N, are independent. Therefore, the events E and F will be independent, if it satisfies these two criteria:

(a) P(N | M) = P(N), provided P (M) ≠ 0

(b) P(M | N) = P(M), provided P (N) ≠ 0

Applying the multiplication theorem on Probability, we can say that P( M ∩ N) = P(M) P(N), then M and N are said to be independent.

The Class 12 Mathematics Chapter 13 notes provide a deep understanding of the concept of independent events for students to score high marks in the exams. 

Consider three events M, N and O. They are said to be mutually independent if the following conditions  are met:

P(M ∩ N) = P(M) P(N)

P(M ∩ O) = P(M) P(O)

P(N ∩ O) = P(N) P(O)

and P(M ∩ N ∩ O) = P(M) P(N) P(O)

Remarks:

• If the two events M and N depend on each other, i.e, if they are not independent, then P(E∩F)≠ P(E). P(F)
• The term “independent” is defined with respect to event probability, and mutually exclusive is defined with respect to the subset of sample space. 
• Mutually exclusive events are defined as a situation when two events do not occur at the same time. Independent event takes place when one event is not affected by the occurrence of the other event.
• Mutually exclusive events will never have shared results, whereas independent events might have shared results.
• Two mutually exclusive events cannot be mutually exclusive, and two mutually exclusive occurrences cannot be mutually exclusive if they have non-zero probabilities of occurrence.
• Two experiments are independent for events E and F, where the first experiment is dependent on the second experiment, then the probability that the events E and F occur simultaneously is given only when both the experiments are performed. It is given as the product of P(E) and P(F), calculated as follows: P(E∩F)=P(E). P(F)

NOTE:

If the two events, E and F, are independent, then we can say that 

(a) E′ and F are independent

(b) E and F’ are independent

(c) E′ and F′ are independent

Students must consider referring to the Chapter 13 Mathematics Class 12 notes for several practice questions based on the Independent events. 

Bayes’ Theorem:

John Bayes is a famous mathematician who used conditional Probability to obtain reverse probability. The ‘Bayes theorem’ was named after him in 1763.

Consider E1, E2,…, En as the partitions associated with sample space. Let E1, E2,…, En be the mutually exclusive and exhaustive events in a sample space. Let A be any event having non-zero probability, then 

P(Ei|A) = P(Ei) P(A|Ei)P(Ei) P(A|Ei)

Applying conditional probability and multiplication rule, P(Ei|A)= P(A ∩ Ei)P(A) = P(Ei) P(A|Ei)P(A)

Now using the theorem of total probability, we get P(Ei|A) = P(Ei) P(A|Ei)P(Ei) P(A|Ei)

Students can refer to the exercise answers and solutions to various problems based on the Bayes Theorem included in the Class 12 Mathematics Chapter 13 notes. 

Explanation:

• Consider two sacs I and II. Let the Sac I contain 22 white and 33 red balls. The Sac II contains 44 white and 55 red balls. One ball must be drawn at random from the sacs.
• The probability of selecting any sac is given as 12, and the probability of drawing a ball of a particular colour from a particular sac. 
• If we know the sac from which the ball is drawn, then the probability that the ball drawn will be of a given colour.
• If the colour of the ball pulled is known, we will find the reverse probability of sac II being selected after the occurrence of an event is known to find the probability that the ball drawn is from a certain sac II.

Remark

When Bayes’ theorem is applied, the nomenclature given below is used:

• Hypotheses are occurrences of events such as E1, E2,…, En
• The probability of the hypotheses Ei is given as P(Ei)
• The posteriori probability of a hypothesis Ei is defined as the conditional Probability P(Ei|A).
• The formula for the occurrence of “causes” is also termed as the “causes formula.” 
• As the Ei’s are a subset of the sample space S, only one of the Ei‘s takes place (that is, only one of the events Ei has to occur). As a result, with reference to the occurrence of event A, the foregoing formula gives us the occurrence of a specific Ei.

Partition of a Sample Space:

The partition of any sample space S is said to be a set of events E1, E2,…, En. 

If Ei ∩ Ej = φ, i ≠ j, i, j = 1, 2, 3, …, n

E1∪ E2 ∪…∪ En= S and P (Ei) > 0 for all i = 1, 2, ……,n.

If the events E1, E2,…, En are disjoint, exhaustive, and have non-zero probabilities, then they represent a partition of the given sample space S.

Consider any nonempty event E. Let its complement be E′. The event E and its complement will form a partition associated with the sample space S as they satisfy E ∩ E′ = φ and E ∪ E′ = S.

Also, for any two events E and F associated with the sample space S, then set {E ∩ F′, E ∩ F, E′ ∩ F, E′ ∩ F′} is also said to be the partition of the sample space S.

The Partition of a sample space is an important concept from the exam point of view. Students can use the Class 12 Mathematics Chapter 13 notes to understand this concept clearly. 

A  Theorem of Total Probability:

Consider {E1, E2,…, En} to be the partition of the sample space S. Suppose that each event E1, E2,…, Enhas a non-zero probability of occurrence. If A is an event linked with sample space S, then the probability of A is given as P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + P(E3) P(A|E3) + ……..+ P(En) P(A|En)

                                = j=1nP(Ej) P(A|Ef)

We know that S = E1∪ E2 ∪…∪ En and Ei ∩ Ej = φ.

For event A, we have, A = A ∩ S = A ∩ (E1∪ E2 ∪…∪ En ) 

Therefore, A = (A ∩ E1) ∪ (A ∩ E2) ∪ (A ∩ E3) ∪……. ∪ (A ∩ En) 

Thus, P(A) = P[(A ∩ E1) ∪ (A ∩ E2) ∪ (A ∩ E3) ∪……. ∪ (A ∩ En)]

                   = P(A ∩ E1) + P (A ∩ E2) + P (A ∩ E3) ……. + P(A ∩ En) 

Applying multiplication rule, we get P(A ∩ E1)= P(E1) P(A|E1)

Therefore, P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + P(E3) P(A|E3) + ……..+ P(En) P(A|En)

                            = j=1nP(Ej) P(A|Ef)

Many questions based on the theorem of total probability are included in the Class 12 mathematics Chapter 13 notes. 

Random Variable’s Probability Distribution:

In the following cases, we were interested in numbers associated with the particular outcome. 

• When two dice are tossed, we may be interested in the total sum of the numbers on the dice.
• When an unbiased coin is tossed n times, we may be interested in the number of heads or tails obtained.

In these examples, we have a rule which assigns a real number to each outcome of an experiment. This real number may be different for different outcomes. Therefore, it is called a variable. Also, the value of this variable depends upon the outcome of the random experiment and, hence, it is called a random variable, denoted by X.

Therefore, a random variable X is a real number and a real-valued function associated with the same sample space of the random experiment.

Probability distribution of any random variable, say X: 

Consider a random variable X. The probability distribution of X is a description which offers the values of random variables and their probabilities.

Therefore, a random variable’s probability distribution can be defined as follows:

Let X be x1, x2,…, xnassociated with probabilities P1, P2,…, Pn, respectively, where pi>0, i=1npi=1 and i=1,2,……,n.

The real numbers x1, x2,…, xn give all the possible values of any random variable X, and pi(i=1,2,…,n) is said to be the probability of a given random variable X by taking the value xi, which means P(X= xi) = pi.

All elements included in the sample space are covered for all values of the random variable X. Therefore, the total probability of X in a probability distribution must be equal to one.

At certain points in the sample, space is X=xi is true.

As a result, the probability that the random variable X takes the value xi is never 0, that is, P(X=xi) ≠ 0

Students can gain practice by solving unlimited questions based on this concept. The mean and variance is an important concept from this chapter which is explained in a stepwise manner in the Class 12 Mathematics Chapter 13 notes. 

Mean of the Random Variable: 

Mean is the measure of a central tendency that is used to roughly locate a middle or an average value of the random variable.

Consider a random variable X, whose possible values are x1, x2,…, xn occur with probabilities P1, P2,…, Pn, respectively. The mean of X, denoted by μ, is defined as the number i=1nxiPi i.e. the mean of X is defined as the weighted average of all the possible values of X as each value is weighted by its probability of occurrence. The mean of a random variable X is also termed as the expectation of X and is denoted by E(X).

Therefore, E(X) = μ = i=1nxiPi = x1P1+ x2P2+ x3P3+……..+ xnPn. 

In order words, the sum of all the products of possible values (xi) by their respective probabilities(Pi) is the mean or  expectation of any random variable X.

Variance of a Random Variable:

• The mean of a random variable X provides no information about the variability of the different values of the random variable.
• If the variance is small, the values of the random variable are near to the mean. Also, the means of random variables X and Y having different probability distributions can be equal.
• To get the difference between X from Y, we require something which measures the extent to which the random variable’s values spread out. 
• Variance is a measure of central tendency which calculates the spread and scatter in the data.

Consider a random variable X, whose possible values are x1, x2,…, xn occur with probabilities P1, P2,…, Pn, respectively. Let E(X) = μ = i=1nxiPi = x1P1+ x2P2+ x3P3+……..+ xnPn be the mean of X. Then, the variance of X, denoted by Var (X) or x2, is given as 

x2= Var (X) = i=1n(xi-μ)2 p(xi)or x2 = E(X-μ)2

As a result, x= i=1n(xi-μ)2 p(xi)is the standard deviation of the random variable X.

Another Formula for Variance:

 Var (X) = i=1n(xi-μ)2 p(xi)= i=1n(xi2+2-2xiμ) p(xi)

Therefore, x2= Var (X) =i=1n(xi2) p(xi)+i=1n2p(xi)–i=1n2xiμ p(xi)

Var (X) = i=1n(xi2) p(xi)+2i=1np(xi)-2μi=1nxi p(xi)

We know that, i=1np(xi)= 1 and i=1nxi p(xi)=

Therefore, Var (X) = i=1n(xi2) p(xi)+2-22 

                                 = i=1n(xi2) p(xi)–2 

Or x2= Var (X) = i=1n(xi2) p(xi)–(i=1nxiPi)2

Or x2= Var (X) = E(X2) – [E(X)]2

Students will get an idea of all relevant formulas, derivations, important questions and concepts included in the Class 12 Mathematics Chapter 13 notes. 

Bernoulli trials:

Consider any dichotomous experiments. For example, when the unbiased coin is tossed, it shows a ‘head’ or a ‘tail’, the response to a question can be ‘yes’ or ‘no’, etc. In such cases, the outcomes are either a ‘success’ and ‘failure’ or ‘not success’.

Here, the outcome of any trial does not depend on the other trial. In such trials, the probability of success or failure does not change, or we can say that the probability remains constant. The independent trials, which can have only two outcomes, such as ‘success’ or ‘failure’, are termed to be the Bernoulli trials. 

The trials of any random experiment are known as Bernoulli trials if they satisfy the parameters listed below: : 

(i) The number of trials should be finite. 

(ii) The trials must not depend on each other (independent). 

(iii) Each trial should have exactly two outcomes: success or failure. 

(iv) The probability of success and failure remains constant in each trial.

The probability of success in any Bernoulli trial is denoted by p, and the probability of failure is denoted by q, such that p + q =1.

The Class 12 Mathematics Chapter 13 notes include various practice questions for students to practice. 

Binomial Distribution: 

The binomial distribution is the expansion of the Bernoulli trials. The binomial expansion for the probabilities of n number of successes in n-Bernoulli trials.  

In the experiment of n-Bernoulli trials, the probabilities of 0, 1, 2,…, n successes are obtained as 1st, 2nd,…,(n + 1)th terms in the expansion of (q + p)n. 

We prove this by finding out the probability of x-successes in an experiment of n-Bernoulli trials.

For x successes, we will have (n-x) failures. 

The x successes (S) and (n – x) failures (F) are obtained in n!x! (n-x)! ways. The probability of successes and failures is = P(x successes). P(n–x) failures = P(S). P(S)…… P(S) (x times). P(F). P(F)……P(F) (n-x) times

Therefore, probability = pxqn-x.

Thus, the probability in a binomial probability distribution is given as P(x successes) = n!x! (n-x)! pxqn-x, where x = 0, 1, 2, 3, ….., n and q = 1 – p 

Therefore, in the binomial expansion (q + p)n, The probability of success is Cxn pxqn-x. The binomial distribution is denoted by B(n, p).

For any value x= r, the probability distribution is given as

P(X=r)= Crn prqn-r 

Visit the links provided below and get the key points and summary in the Class 12 Mathematics Chapter 13 notes associated with the CBSE syllabus.

Class 12 Mathematics Chapter 13 Notes: Exercise &  Solutions

The Class 12 Mathematics Chapter 13 notes provide a strong conceptual understanding of the topics included in the NCERT books. It covers all important questions, formulas, and theorems with stepwise and detailed explanations. They also provide unlimited practice questions. The Class 12 Mathematics Chapter 13 notes will help students in the effective preparation for the Board exams as well as the undergraduate entrance exams such as JEE, BITSAT, etc. 

Extramarks, an online learning platform, brings the Class 12 Mathematics Chapter 13 notes, prepared by our subject matter experts  by analysing several CBSE  past years’ question papers to facilitate precise understanding of all complex concepts. These NCERT Solutions include well explained and step-by-step explanations of all the problems given in the CBSE books. 

Besides the Class 12 Mathematics Chapter 13 notes, Extramarks also provides Exemplar notes, CBSE Sample papers, MCQ worksheets, CBSE revision notes and mock tests as academic materials for students. 

Click on the links mentioned below to get access to Class 12 Mathematics Chapter 13 notes to excel in examinations. 

Class 12 Mathematics Chapter 13 Notes: Key Features

The Class 12 Mathematics Chapter 13 notes include

• Class 12 Mathematics Chapter 13 notes are a vital tool for students in the home assignments and examination preparation.
• The solutions are regarded as the best support as it includes in-depth solutions to all problems. 
• Several solved examples are  provided  in the notes for a clear understanding of all concepts.
• The Class 12 Mathematics Chapter 13 notes can be accessed through any electronic device such as a mobile, laptop, tab, etc. 
• It is detailed,  precise and well structured. 
• With the help of the notes, students can save time as they get apt  information needed for the board exams and get enough time for other subjects as well. 
• The Class 12 Mathematics Chapter 13 notes are very helpful for quick revisions. 

Preparation Tips For Class 12 Board Exams:

1. Refer to NCERT Books: Students are advised to use the NCERT books for preparing for the board exams. The Class 12 Mathematics Chapter 13 notes can also be studied along with the NCERT textbook.
2. Know the exam details: Before starting with the preparation, students must know about the paper pattern, marking system and weightage of each chapter. 
3. Prepare a timetable: Students appearing for Class 12 Board exams are advised to plan their schedule and follow it strictly. This will allow them to cover the entire CBSE syllabus.
4. Use the best notes: Students can consider using the Class 12 Mathematics chapter 13 notes as they are completely  reliable  and include in-depth information about all concepts.
5. Appear for tests: Students can appear for several practice tests and mock tests available on   Extramarks website. Practice the CBSE extra questions given in the Class 12 Mathematics notes Chapter 13 until you get all the answers correct.