Bayes Theorem | Definition, Proof, Formula, Derivation & Solved Examples | Class 12 Math Notes Study Material Download Free PDF

Bayes Theorem Statement

Let E1, E2,…, En be a set of events associated with a sample space S, where all the events E1, E2,…, En have nonzero probability of occurrence and they form a partition of S. Let A be any event associated with S, then according to Bayes theorem,

\(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i)P(A│E_i)}{\sum\limits_{k=1}^{n}P(E_k)P(A| E_k)}\end{array} \)

for any k = 1, 2, 3, …., n

Bayes Theorem Proof

According to the conditional probability formula,

\(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i ∩ A)}{P(A)}…(1)\end{array} \)

Using the multiplication rule of probability,

\(\begin{array}{l}P(E_i ∩ A)~= ~P(E_i)P(A │E_i)…(2)\end{array} \)

Using total probability theorem,

\(\begin{array}{l}P(A)~=~\sum\limits_{k=1}^{n}~P(E_k)P(A| E_k)…(3)\end{array} \)

Putting the values from equations (2) and (3) in equation 1, we get

\(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i)P(A│E_i)}{\sum\limits_{k=1}^n~P(E_k)P(A| E_k)}\end{array} \)

Note:

The following terminologies are also used when the Bayes theorem is applied:

Hypotheses: The events E1, E2,… En is called the hypotheses

Priori Probability: The probability P(Ei) is considered as the priori probability of hypothesis Ei

Posteriori Probability: The probability P(Ei|A) is considered as the posteriori probability of hypothesis Ei
Bayes’ theorem is also called the formula for the probability of “causes”. Since the Ei‘s are a partition of the sample space S, one and only one of the events Ei occurs (i.e. one of the events Ei must occur and the only one can occur). Hence, the above formula gives us the probability of a particular Ei (i.e. a “Cause”), given that the event A has occurred.

Bayes Theorem Formula

If A and B are two events, then the formula for the Bayes theorem is given by:

\(\begin{array}{l}P(A|B)= \frac{P(B|A)P(A)}{P(B)}\:\:where\:\:P(B)\neq 0\end{array} \)  

Where P(A|B) is the probability of condition when event A is occurring while event B has already occurred.

Bayes Theorem Derivation

Bayes Theorem Derivation

Bayes Theorem can be derived for events and random variables separately using the definition of conditional probability and density.

From the definition of conditional probability, Bayes theorem can be derived for events as given below:

P(A|B) = P(A ⋂ B)/ P(B), where P(B) ≠ 0

P(B|A) = P(B ⋂ A)/ P(A), where P(A) ≠ 0

Here, the joint probability P(A ⋂ B) of both events A and B being true such that,

P(B ⋂ A) = P(A ⋂ B)

P(A ⋂ B) = P(A | B) P(B) = P(B | A) P(A)

P(A|B) = [P(B|A) P(A)]/ P(B), where P(B) ≠ 0

Similarly, from the definition of conditional density, Bayes theorem can be derived for two continuous random variables namely X and Y as given below:

\(\begin{array}{l}f_{X|Y=y}(x)=\frac{f_{X,Y(x,y)}}{f_Y(y)}\\f_{Y|X=x}(y)=\frac{f_{X,Y(x,y)}}{f_X(x)}\end{array} \)

Therefore, 

\(\begin{array}{l}f_{X|Y=y}(x)=\frac{f_{Y|X=x}(y)f_X(x)}{f_Y(y)}\end{array} \)

Examples and Solutions

Some illustrations will improve the understanding of the concept.

Example 1:

A bag I contains 4 white and 6 black balls while another Bag II contains 4 white and 3 black balls. One ball is drawn at random from one of the bags, and it is found to be black. Find the probability that it was drawn from Bag I.

Solution:

Let E1 be the event of choosing bag I, E2 the event of choosing bag II, and A be the event of drawing a black ball.

Then,

\(\begin{array}{l}P(E_1)~ = ~P(E_2)~ =~\frac{1}{2}\end{array} \)

Also, P(A|E1) = P(drawing a black ball from Bag I) = 6/10 = 3/5

P(A|E2) = P(drawing a black ball from Bag II) = 3/7

By using Bayes’ theorem, the probability of drawing a black ball from bag I out of two bags,

\(\begin{array}{l}P(E_1 |A)~ =~\frac{P(E_1)P(A|E_1)}{P(E_1 )P(A│E_1 )+ P(E_2)P(A|E_2)}\end{array} \)

\(\begin{array}{l}=\large\frac{\frac{1}{2}~\times~\frac{3}{5}}{\frac{1}{2}~\times~\frac{3}{5}~+~\frac{1}{2}~ ×~\frac{3}{7}}\end{array} \)

\(\begin{array}{l}=\frac{7}{12}\end{array} \)

Example 2:

A man is known to speak the truth 2 out of 3 times. He throws a die and reports that the number obtained is a four. Find the probability that the number obtained is actually a four.

Solution:

Let A be the event that the man reports that number four is obtained.

Let E1 be the event that four is obtained and E2 be its complementary event.

Then, P(E1) = Probability that four occurs = 1/6.

P(E2) = Probability that four does not occur = 1- P(E1) = 1 – (1/6) = 5/6.

Also, P(A|E1)= Probability that man reports four and it is actually a four = 2/3

P(A|E2) = Probability that man reports four and it is not a four = 1/3.

By using Bayes’ theorem, probability that number obtained is actually a four, P(E1|A)

\(\begin{array}{l}= \large \frac{P(E_1)P(A|E_1)}{P(E_1 )P(A│E_1 )~+~ P(E_2)P(A|E_2)}~
=~\frac{\frac{1}{6} ~ ×~ \frac{2}{3}}{\frac{1}{6} ~×~ \frac{2}{3}~ +~ \frac{5}{6}~ ×~\frac{1}{3}}\end{array} \)

\(\begin{array}{l}=\frac{2}{7}\end{array} \)

Bayes Theorem Applications

One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. Bayesian inference has found application in various activities, including medicine, science, philosophy, engineering, sports, law, etc. For example, we can use Bayes’ theorem to define the accuracy of medical test results by considering how likely any given person is to have a disease and the test’s overall accuracy. Bayes’ theorem relies on consolidating prior probability distributions to generate posterior probabilities. In Bayesian statistical inference, prior probability is the probability of an event before new data is collected.

Practice Problems

Solve the following problems using Bayes Theorem.

  1. A bag contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted, and again the ball is returned to the bag. Also, 2 additional balls of the colour drawn are put in the bag. After that, the ball is drawn at random from the bag. What is the probability that the second ball drawn from the bag is red?
  2. Of the students in the college, 60% of the students reside in the hostel and 40% of the students are day scholars. Previous year results report that 30% of all students who stay in the hostel scored A Grade and 20% of day scholars scored A grade. At the end of the year, one student is chosen at random and found that he/she has an A grade. What is the probability that the student is a hosteler? 
  3. From the pack of 52 cards, one card is lost. From the remaining cards of a pack, two cards are drawn and both are found to be diamond cards. What is the probability that the lost card is a diamond?

Frequently Asked Questions on Bayes Theorem

Q1

What is meant by Bayes theorem in probability?

In Probability, Bayes theorem is a mathematical formula, which is used to determine the conditional probability of the given event. Conditional probability is defined as the likelihood that an event will occur, based on the occurrence of a previous outcome.

Q2

How is Bayes theorem different from conditional probability?

As we know, Bayes theorem defines the probability of an event based on the prior knowledge of the conditions related to the event. In case, if we know the conditional probability, we can easily find the reverse probabilities using the Bayes theorem.

Q3

When can we use Bayes theorem?

Bayes theorem is used to find the reverse probabilities if we know the conditional probability of an event.

Q4

What is the formula for Bayes theorem?

The formula for Bayes theorem is:
P(A|B)= [P(B|A). P(A)]/P(B)
Where P(A) and P(B) are the probabilities of events A and B.
P(A|B) is the probability of event A given B
P(B|A) is the probability of event B given A.

Q5

Where can we use Bayes theorem?

Bayes rule can be used in the condition while answering the probabilistic queries conditioned on the piece of evidence.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Max Marks: 80

No.UnitsMarks
I.Relations and Functions08
II.Algebra10
III.Calculus35
IV.Vectors and Three – Dimensional Geometry14
V.Linear Programming05
VI.Probability08
Total Theory80
Internal Assessment20
Grand Total100

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals 

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Q1

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.

Q2

How many units are discussed in the CBSE Class 12 Maths Syllabus?

In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.

Q3

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.