Class – 11

Chapter – 1 Sets

Set: A set is a well-defined collection of objects.

Sets Representation: Sets can be represented in many ways. They differ in the way in which the
elements are listed. The three ways of representing sets are 'Semantic form', 'Roster form', 'Set
builder form'.

Semantic Form: The semantic form describes a statement to show what are the elements of a set.

Roster Form: The most common form used to represent sets is the roster form in which the
elements of the sets are enclosed by curly brackets separated by commas.
Finite Roster Form Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers)
Infinite Roster Form Notation of Sets : Set B = {5, 10, 15, 20 ....} (The multiples of 5)

Set Builder Form: Set builder form has a certain rule or a statement that specifically describes the
common feature of all the elements of a set. The set builder form uses a vertical bar in its
representation, with a text describing the character of the elements of the set.

Types of Sets

There are many types of sets based on the number of elements in a set. They are as follows.

Singleton Sets: A set that has only one element is called a singleton set or also called a unit set.
Example, Set A = { k | k is a number between 3 and 5 } which is A = {4}.
Finite Sets: As the name implies, a set with a finite or countable number of elements is called a finite
set.
Example, Set B = { k | k is set of prime numbers within 20}, which is B = {2,3,5,7,11,13,17,19}
Infinite Sets: A set with an infinite number of elements is called an infinite set.
Example: Set C = {Multiples of 3}, which is {3,6,9,12,15 ..........}
Empty or Null Sets: A set that does not contain any elements is called an empty set or a null set. An
empty set is denoted using the symbol '∅'. It is read as 'phi'.
Example: Set X = {}
Equal Sets: If two sets have the same elements in them, then they are called equal sets.
Example: A = {1,2,3} B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A =
B.
Unequal Sets: If two sets have at least one element that is different, then they are unequal sets.
Example: A = {1,2,3} B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as
A ≠ B.
Equivalent Sets: Two sets are said to be equivalent sets when they have the same number of
elements, though the elements are different.
Example: A = {1,2,3,4} B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)
Overlapping Sets: Two sets are said to be overlapping if at least one element from set A is present in
set B.
Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A
and B are overlapping sets.
Disjoint Sets: Two sets are disjoint sets if there are no common elements in both sets.
Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.
Subset and Superset: For two sets A and B, if every element in set A is present in set B, then set A is
a subset of set B and B is the superset of set A.
Example: A = {1,2,3} B = {1,2,3,4,5,6}
A ⊆ B, since all the elements in set A are present in set B.
B ⊇ A denotes that set B is the superset of set A.
Universal Set: Universal set is the collection of all the elements in regards to a particular subject. The
universal set is denoted by the letter 'U'.
Example: Let U = {The list of all road transport vehicles}. Here, a set of cars are a subset for this
universal set, the set of cycles, trains are all subsets of this universal set.
Power Sets: Power sets contain the set of all subsets that a set could contain.

Operations on Sets:

• Union of sets, which is denoted as A U B, lists the elements in set A and set B
or both the elements in set A and set B.
• The intersection of sets which is denoted by A ∩ B lists the elements that are
common to both set A and set B.
• Set Difference which is denoted by A - B, lists the elements in set A that are not
present in set B.
• Set Complement which is denoted by A', is the set of all elements in the
universal set that are not present in set A. In other words, A' is denoted as U - A,
which is the difference in the elements of the universal set and set A.
• Cartesian Product of two sets which are denoted by A×B, is the product of two
non-empty sets, wherein ordered pairs of elements are obtained.
Sets Formulas

Sets find their application in the field of algebra, statistics, and probability. There are some
important formulas related to sets. They are as follows.
For any two overlapping sets A and B, then,

• n(A U B) = n(A) + n(B) - n(A ∩ B)

For any two sets A and B that are disjoint,
• n(A U B) = n(A) + n(B)
• n (A ∩ B) = n(A) + n(B) - n(A U B)
• n(A) = n(A U B) + n(A ∩ B) - n(B)
• n(B) = n(A U B) + n(A ∩ B) - n(A)
• n(A - B) = n(A U B) - n(B)
• n(A - B) = n(A) - n(A ∩ B)

Sets Notation Symbols

There are some symbols used in sets, which is very important while reading them. The most
commonly used notations in sets are tabulated below.
 Notation Meaning

U Universal Set

n(X) Cardinal number of set X

b∈A 'b' is an element of set A

a∉B 'a' is not an element of set B

{} Denotes a set

∅ Null or empty set

AUB Set A union set B

A∩B Set A intersection set B

A⊆B Set A is a subset of set B

B⊇A Set B is the superset of set A

Sets Properties

Similar to numbers sets also have properties like associative property, commutative
property, and so on. There are six important properties of sets. Given, three sets A, B, and
C, the properties for these sets are as follows.

Name Property

AUB=BUA
Commutative Property
A∩B=B∩A
 Name Property

(A ∩ B) ∩ C = A ∩ (B ∩ C)
Associative Property
(A U B) U C = A U (B U C)

A U (B ∩ C) = (A U B) ∩ (A U C)
Distributive Property
A ∩ (B U C) = (A ∩ B) U (A ∩ C)

AU∅=A
Identity Property
A∩U =A

Complement Property A U A' = U

A∩A=A
Idempotent Property
AUA=A

Chapter – 2 Relations and Functions

Function means a correspondence from one value x of the first set to another value y of the second
set.

Domain is defined as the set of all the values that the function can input while it can be defined.

Range are all the values that come out as output of the function involved.

Codomain is the set of values that have the potential of coming out as outputs of a function.

Types of Functions

There are many different types of functions that we encounter in mathematics. We will define them
here:

1. One-One functions

Also called Injective functions, are the functions where each input is connected to a unique output.
If x1 is connected to y1, then there will be no other input x in the domain which will be connected to
y1.

2. Many-One function
It is the opposite of One-One functions. In this type, more than one input x’s can be connected to
the same y.

3. Onto functions

Also called Surjective functions, they are the functions where each of the y that exist in the
codomain is connected to x.

4. Into functions

5. They can contain some values in codomain which are not connected to the input set.

A function that is both one-one and onto is called Bijective functions. If we are told to find the
inverse of a function, the function must be bijective otherwise it is not possible to find the inverse.

Some functions that are commonly encountered are:

1. Constant function

There is only one output value for all the input values of x. Their graph appears as a straight line
parallel to the x-axis.

2. Polynomial functions

These functions look like as shown below

f(x)=anxn+an−1xn−1+⋯ + a0,

The highest power of x is called the degree of the polynomial. Here, an, an−1, ................, a1, a0 are the
coefficients of respective powers of x. The linear function and quadratic functions are special cases
of polynomial functions.

3. Linear functions

These functions are just a linear combination of variables with coefficients. For example, y = 2x + 3 is
a linear function. They appear as straight lines in the x-y axis graph.

4. Trigonometric functions

These functions are composed of various trigonometric ratios.

There are many other types of functions as well that we have not touched upon in this blog. Some
are very complex that you will encounter later.

Types of Relations

Let us define a set A. We can now form a relation R from A to A. Let A = {1,2} and now, let us look at
different types of relations that can be:

1. Universal Relation
If all elements of A are related to every element of A, then it is a universal relation. For example,the
universal relation R on A will be, {(1,1), (2,2), (1,2), (2,1)}.

2. Reflexive Relation

For every element ‘a’ in set A, if (a, a) exists in the relation R, then R is reflexive. For example, if (1,1)
and (2,2) are both part of relation R, hence, R is reflexive.

3. Symmetric Relation

For every ordered pair (a,b) in R, if the pair (b, a) also exists in R, then R is symmetric. If (1,2) and
(2,1) are both present in R, it is symmetric.

4. Transitive Relation

For every (a,b) and (b,c) that exist in R, if (a,c) is present then R is transitive.

5. Equivalence Relation

If a relation R is reflexive, symmetric, and transitive, then R is an equivalence relation.

6. Empty Relation

If none of the elements of set A are related to other elements of set A, then R is empty.

Chapter – 3 Trigonometric Functions

1. Basic Formulas

• sin θ = Opposite Side/Hypotenuse

• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

2. Reciprocal Identities

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

3. Trigonometric Ratio Table

4. Periodic Identities (in Radians)

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A

• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A & cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

5. Co-function Identities (in Degrees)

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

6. Sum and Difference Identities

The sum and difference identities include the formulas of sin(x+y), cos(x-y), cot(x+y), etc.

• sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

• cos(x+y) = cos(x)cos(y) – sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x • tan y)
• sin(x–y) = sin(x)cos(y) – cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)
7. Half Angle Identities

8. Double Angle Identities

The double of the angle x is presented through the below few formulae.

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]

• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
• tan(2x) = [2tan(x)]/ [1−tan2(x)]
• sec (2x) = sec2 x/(2-sec2 x)
• cosec (2x) = (sec x. cosec x)/2

9. Triple Angle Identities

• Sin 3x = 3sin x – 4sin3x

• Cos 3x = 4cos3x-3cos x
• Tan 3x = [3tanx-tan3x]/[1-3tan2x]

10. Product identities

• sinx⋅cosy=sin(x+y)+sin(x−y)2
• cosx⋅cosy=cos(x+y)+cos(x−y)2
• sinx⋅siny=cos(x−y)−cos(x+y)2

11. Sum to Product Identities

The combination of two acute angles A and B can be presented through the trigonometric
ratios, in the below formulae.

• sinx+siny=2sinx+y2cosx−y2
• sinx−siny=2cosx+y2sinx−y2
• cosx+cosy=2cosx+y2cosx−y2
• cosx−cosy=−2sinx+y2sinx−y2

Chapter – 4 Principal of Mathematical Induction

Statement: A sentence is called statement if either it is true or false.

Mathematical induction is the process of proving a mathematical statement (or) an equation (or) a
theorem for the set of all natural numbers, NN.

Steps of Mathematical Induction: There are three steps and each step is named as follows:

Chapter – 5 Complex Numbers and Quadratic Equations

Complex number is the sum of a real number and an imaginary number.

A complex number is of the form a + ib and is usually represented by z. Here both a and b are real
numbers.
The value 'a' is called the real part which is denoted by Re(z), and 'b' is called the imaginary part
Im(z). Also, ib is called an imaginary number.

Iota is represented by i and we have the value of i2 = -1

Graphing of Complex Numbers

Argand Plane: The Euclidean plane with reference to complex numbers is called the complex plane
or the Argand Plane

The complex number z = a + ib is represented with the real part - a, with reference to the x-axis, and
the imaginary part-ib, with reference to the y-axis.
Modulus of the Complex Number: The distance of the complex number represented as a
point in the argand plane (a, ib) is called the modulus of the complex number.
This distance is a linear distance from the origin (0, 0) to the point (a, ib), and is measured
as

Argument of the Complex Number: The angle made by the line joining the geometric
representation of the complex number and the origin, with the positive x-axis, in the
anticlockwise direction is called the argument of the complex number.
Argz (θ) = Tan-1b/a

Polar Representation of a Complex Number: With the modulus and argument of a

complex number and the representation of the complex number in the argand plane.
z = r(Cosθ + iSinθ).

Properties of Complex Numbers: The following properties of complex numbers are helpful to
better understand complex numbers

• Conjugate of a Complex Number: The conjugate of the complex number is formed

by taking the same real part of the complex number, and changing the imaginary part
of the complex number to its additive inverse. For a complex number z = a + ib, its
conjugate is z¯ = a - ib.
• Reciprocal of a complex Number: The reciprocal of complex numbers is helpful in
the process of dividing one complex number with another complex number. The
reciprocal of the complex number z = a + ib is

• Equality of Complex numbers: Two complex numbers are said to be equal if the
real part of both the complex numbers are equal a1=a2, and the imaginary parts of
both the complex numbers are equal b1=b2.
• Ordering of Complex Numbers: The ordering of complex numbers is not possible.
Real numbers and other related number systems can be ordered, but complex
numbers cannot be ordered.
• Euler’s Formula: As per Euler's formula and for the functional representation of x
and y we have ex + iy = ex(cosy + isiny) = excosy + iexSiny.

Operations on Complex Numbers

Addition: Here in complex numbers, the real part is added to the real part and the imaginary
part is added to the imaginary part. The complex numbers follow all the following properties
of addition.

Subtraction: Here for any two complex numbers, the subtraction is separately performed
across the real part and then the subtraction is performed across the imaginary part.

Multiplication: Here the absolute values of the two complex numbers are multiplied and their
arguments are added to obtain the product of the complex numbers.

Algebraic Identities of Complex Numbers:

Chapter – 6 Linear Inequalities

Linear inequalities are defined as expressions in which two values are compared using the
inequality symbols. The five symbols that are used to represent inequality are listed below:

Symbol Name Symbol Example

Not equal ≠ x≠3

Less than (<) x + 7 < √2

Greater than (>) 1 + 10x > 2 + 16x

Less than or equal to (≤) y≤4

Greater than or equal to (≥) -3 - √3x ≥ 10

Rules of Inequalities

The 4 types of operations that are done on linear inequalities are addition, subtraction,
multiplication, and division. Inequalities with the same solution are called equivalent inequality.
There are rules for both equality and inequality.
 • Addition Rule of Inequality: As per the addition rule of inequality, adding the same number
to each side of the inequality produces an equivalent inequality
If x > y, then x + a > y + a and if x < y, then x + a < y + a.
• Subtraction Rule of Inequality: As per the subtraction rule of inequality, subtracting the
same number from each side of the inequality produces an equivalent inequality,
If x > y, then x − a > y − a and if x < y, then x − a < y − a.
• Multiplication Rule of Inequality: As per the multiplication rule of inequality, multiplication
on both sides of an inequality with a positive number always produces an equivalent
inequality
If x > y and a > 0, then x * a > y * a and if x < y and a > 0, then x * a < y * a, Here, * is used as
the multiplication symbol.
On the other hand, multiplication on both sides of an inequality with a negative number
does not produce an equivalent inequality unless we also reverse the direction of the
inequality symbol.
If x > y and a < 0, then x * a < y * a and if x < y and a < 0, then x * a > y * a.
• Division Rule of Inequality: As per the division rule of inequality, division of both sides of an
inequality with a positive number produces an equivalent inequality,
If x > y and a > 0, then (x/a) > (y/a) and if x < y and a > 0, then (x/a) < (y/a).
On the other hand, the division of both sides of an inequality with a negative number
produces an equivalent inequality if the inequality symbol is reversed.
If x > y and a < 0, then (x/a) < (y/a) and if x < y and a < 0, then (x/a) > (y/a)

Solving Linear Equalities: Solving a multi-step linear inequality is the same as solving multi-
step linear equations; begin by isolating the variable from the constants.

Step 1: Simplify the inequality on both sides - on LHS as well as RHS as per the rules of inequality.

Step 2: When the value is obtained, if the inequality is a strict inequality, the solution for x is less
than or greater than the value obtained as defined in the question. And, if the inequality is not a
strict inequality, then the solution for x is less than or equal to or greater than or equal to the value
obtained as defined in the question.

Important Notes on Linear Inequalities

Here is a list of a few points that should be remembered while studying linear inequalities:

• In the case of inequality, some other relationship like less than or greater than exists
between LHS and RHS.

• A linear inequality is called so due to the highest power of the variable being 1.

• "Less than" and "greater than" are strict inequalities while "less than or equal to" and
"greater than or equal to" are not strict inequalities.

• For every linear inequality which uses strict inequality, the value obtained for x is shown by a
hollow dot. It shows that the value obtained is excluded.
 • For every inequality which is not strict inequality, the value obtained for x is shown by a solid
dot. It shows that the value obtained is included.

Chapter – 7 Permutations and Combinations

Permutation: We can associate the word permutations with the word arrangements. When you
read the phrase “number of permutations”, think of the phrase “number of arrangements”.

Types of Permutations:

Combinations: Combinations correspond to the selection of things (and not their

arrangement). We do not intend to arrange things. We intend to select them.
 Chapter – 8 Binomial Theorem

Binomial theorem used to find the powers of the binomials which cannot be expanded using
the algebraic identities.

Chapter – 9 Sequences and Series

Arithmetic Progression: We can define an arithmetic progression in two ways:

An arithmetic progression is a sequence where the differences between every two consecutive
terms are the same.

An arithmetic progression is a sequence where each term, except the first term, is obtained by
adding a fixed number to its previous term.

Terms Used in Arithmetic Progression:

First Term: As the name suggests, the first term of an AP is the first number of the progression. It is
usually represented by a1 (or) a.

Common Difference: We know that an AP is a sequence where each term, except the first term, is
obtained by adding a fixed number to its previous term. Here, the “fixed number” is called the
“common difference” and is denoted by 'd'.

General Term of Arithmetic Progression (Nth Term): The general term (or) nth term of an AP whose
first term is a and the common difference is d is found by the formula:
an = a + (n-1)d.

Formula for Calculating Sum of Arithmetic Progression: Consider an arithmetic progression (AP)
whose first term is a1 (or) a and the common difference is d.

The sum of first n terms of an arithmetic progression when the nth term is NOT known is
Sn = n/2[2a+(n-1) d]

The sum of first n terms of an arithmetic progression when the nth term, an is known is
Sn = n/2[a1+an].

Geometric Progressions: A geometric progression is a sequence where every term bears a constant
ratio to its preceding term.

Types of Geometric Progressions: The two types of a geometric progression are the finite
geometric progression and the infinite geometric progression. The details of the two
geometric progressions are as follows.

Finite geometric progression: Finite geometric progression is the geometric series that
contains a finite number of terms. It is the sequence where the last term is defined.

Infinite geometric progression: Infinite geometric progression is the geometric series that
contains an infinite number of terms. It is the sequence where the last term is not defined.

nth term = The formula for the nth term of the geometric progression is: an = arn-1

Sum of Series:

• For finite series: Sn = a(1−rn)/(1−r) for r≠1, and Sn = an for r = 1

• For infinite series:
o Case 1: When |r| < 1 S∞ = a/(1 - r)
o Case 2: When |r| > 1 S∞ = ∞

Chapter – 10 Straight Lines

We require two perpendicular axes to locate a point in the plane. One of them is horizontal and
other is Vertical

The plane is called Cartesian plane and axis are called the coordinates axis

The horizontal axis is called x-axis and Vertical axis is called Y-axis
The point of intersection of axis is called origin.

The distance of a point from y axis is called x –coordinate or abscissa and the distance of the point
from x –axis is called y – coordinate or Ordinate

The distance of a point from y axis is called x –coordinate or abscissa and the distance of the point
from x –axis is called y – coordinate or Ordinate

The Origin has zero distance from both x-axis and y-axis so that its abscissa and ordinate both are
zero. So the coordinate of the origin is (0, 0)

A point on the x –axis has zero distance from x-axis so coordinate of any point on the x-axis will be
(x, 0)

A point on the y –axis has zero distance from y-axis so coordinate of any point on the y-axis will be
(0, y)

The axes divide the Cartesian plane in to four parts. These Four parts are called the quadrants

The coordinates of the points in the four quadrants will have sign according to the below table

Important Formulas:

Distance Formula:

Slope:

Section Formula:
Area of a Triangle:

The equation of a line can be found through various methods depending on the available
information. Some of the methods are:

• Point-slope form: This formula is used only when we know the slope of the line and a point
on the line. The equation is: y – y1 = m (x – x1)

Other Important Formulas:

Chapter – 11 Conic Sections

Circles

Parabolas

• Parabola: Locus of all the points equidistant from a fixed line and a fixed point on a
plane.
 • General Equation: The equation of parabola with focus at A(a, 0) and directrix x = -a
is: y2 = 4ax

Ellipse
Chapter – 12 Introduction to three-dimensional geometry
Chapter – 13 Limits and Derivatives

Limits: A limit is defined as a value that a function approaches the output for the given input
values.
Properties of limits:
Chapter – 14 Mathematical Reasoning
 Chapter – 15 Statistics

Statistics - A broad mathematical discipline which studies ways to collect, summarize, and
draw conclusions from data

Data - A systematic record of facts or different values of a quantity is called data. Data is of
two types:

· Primary Data: The data collected by a researcher with a specific purpose in mind is called
primary data.

· Secondary Data: The data gathered from a source where it already exists is called
secondary data

Features of data

· Statistics deals with collection, presentation, analysis and interpretation of numerical data.

· Arranging data in an order to study their salient features is called presentation of data.

· Data arranged in ascending or descending order is called arrayed data or an array

· Range of the data is the difference between the maximum and the minimum values of the
observations
· Table that shows the frequency of different values in the given data is called a frequency
distribution table

· A frequency distribution table that shows the frequency of each individual value in the
given data is called an ungrouped frequency distribution table.

· A table that shows the frequency of groups of values in the given data is called a grouped
frequency distribution table

· The groupings used to group the values in given data are called classes or class- intervals.
The number of values that each class contains is called the class size or class width.

· The lower value in a class is called the lower-class limit.

· The higher value in a class is called the upper-class limit.

· Class mark of a class is the mid value of the two limits of that class.

· A frequency distribution in which the upper limit of one class differs from the lower limit of
the succeeding class is called an Inclusive or discontinuous Frequency Distribution

· A frequency distribution in which the upper limit of one class coincides from the lower limit
of the succeeding class is called an exclusive or continuous Frequency Distribution

Different Types of Graphs

S.No. Types Description

1. Bar Graph Pictorial representation of data in which rectangular-shaped bars

of uniform width are drawn with equal spacing between them on
<image> the x-axis and the value of the variable is shown on the other axis,
that is the y axis.

2. Histogram A type of bar diagram, with class intervals on the horizontal axis
and the frequency of the class interval on the vertical axis. Also,
<image> there is no gap between the bars.

3. Circle Graph Pictorial representation of data that shows the relationship

or Pie-chart between a whole and its part.

<image>

4. Pictograph Representation of data using different images or symbols.

<image>
 5. Line Graph Special type of graph used to display change over time as a
series of data points connected by straight line segments on two
<image> axes.

6. Scatter Plot A scatter plot represents the relationship between two variables.

<image>

Mean: The arithmetic mean of a given data is the sum of all observations divided by the
number of observations

Mean = Sum of observations / Total number of observations

Median: The value of the middlemost observation, obtained after arranging the data in
ascending order, is called the median of the data.

If number of observations is odd then Median = [(n+1) / 2]th observation

If number of observations is even then Median = [(n/2)th observation + ((n+1)/2)th

observation] / 2

Mode: The value which appears most often in the given data i.e. the observation with the
highest frequency is called mode of data
Chapter – 16 Probability

Probability can be defined as the ratio of the number of favourable outcomes to the total number of
outcomes of an event.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space. For
example, the sample space of tossing a coin are head and tail.

Favourable Outcome: An event that has produced the desired result or expected event is called a

favourable outcome. For example, when we roll two dice, the possibility of getting number 4 is (1,3),
(2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Event: The total number of outcomes of a random experiment is called an event.

Probability of an event P(E) = Number of favourable outcomes / Sample Space

Important Notes:

1. Probability is a measure of how likely an event is to happen.

2. Probability is represented as a fraction and always lies between 0 and 1
3. An event can be defined as a subset of sample space.

Conditional probability: It is the probability for one event to occur with some relationship to
one or more other events.
Properties of conditional probability –

(i) If F be an event of a sample space s of an experiment, then P (S/F) = P (F/F) = 1 If A and

B are any two events of a sample space S and F is an event of s such that P (F) ≠ 0, then

(ii) P (A ∩ B/F) = P(A/F) + P (B/F) – P (A ∩ B/F) IF A and Bare disjoint event then P(A ∩ B/F)
= P(A/F) + P (B/F)

(iii) P (E / F) = 1 – P (E/F) or P (E∩/F) = 1 – P (E/F)

Multiplication Theorem On Probability – Let E and F be two events associated with a

sample space S. P(E ∩ F) denotes the probability of the event that both E and F occur,
which is given by P (E ∩F) = P (E) P (F/E) = P (F) P (E/F), provided P (E) ≠ 0 and P(F) ≠ 0

Independent Event–

(i) Events E and F are independent if P (E ≠ F) = P (E) × P (F)

(ii) Two events E and F are said to be independent if P (E/F) = P (E) and P(F/E) = P(F)
provided P (E) ≠ 0 and P (F) 0

(iii) Three events E, F and G are said to be independent or mutually independent if P(E ∩F
∩G) = P(E) P(F) P(G).

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

Bayes’ formula: It describes the probability of occurrence of an event in relation to any

condition. It is useful for the case of conditional probability.