Introduction to Business Statistics

UNIT 1: INTRODUCTION TO PROBABILITY
& STATISTICS
Mr.T.SOMASUNDARAM
ASSISTANT PROFESSOR
DEPARTMENT OF MANAGEMENT
KRISTU JAYANTI COLLEGE, BANGALORE

UNIT 1: INTRODUCTION TO PROBABILITY
& STATISTICS
Definition, Functions, Scope, Limitations
of Statistics, Diagrams and Graphs, Basic
definitions and rules for probability,
conditional probability and independence
of events.

Statistics - Introduction:
The word statistics seems to have been derived from the
Latin word ‘Status’ or Italian word ‘Statista’ or German
word ‘Statistik’ or French word ‘Statistique’, each of which
means a political state.
In ancient period, kings or ruling chiefs used to take
censuses of population and property within their domain to
determine man power and wealth.
In 18th century, mathematics was introduced in the
collection, classification and presentation of data.

Statistics - Introduction:
Nowadays modern science of statistics is extended its scope
to number of department of human knowledge applied to all
fields of enquiry, where a study of large numbers is
involved.
 Statistics has originated as a science of statehood and found
applications in many fields like Agriculture, Economics,
Commerce, Medicine, Industry, planning, education, etc.
It is concerned with scientific methods for collecting,
organizing, summarizing, presenting, analyzing data,
conclusions and making decisions on basis of analysis.

Statistics - Meaning:
 The word ‘Statistics’ is used to refer to numerical facts, such
as the number of people living in particular area.
 The study of ways of collecting, analyzing and interpreting
the facts.
 It refers to statistical methods and principles for classification
and analysis of quantitative data.
 It is a purpose of observing, recording, describing and
enumerating the quantitative data.
 Its purpose is to obtain and explore knowledge.
 It is a body of methods for obtaining information.

Statistics - Definition:
“Statistics is the science which deals with the methods of
collecting, classifying, presenting, comparing and interpreting
numerical data collected to throw some light on any sphere of
enquiry.” - Seligman
“Statistics may be defined as the science of collection,
presentation, analysis and interpretation of numerical data.”
- F.E. Croxton & D.J. Cowden
“Statistics is the science and art of handling aggregate of
facts – observing, enumerating, recording, classifying and
otherwise systematically treating them.” - Harlow

Division of Statistics:
Statistics are classified into two main divisions –
1. Statistical methods – formulation of general rules and
principles applicable in handling different branches of data –
collection, classification, organizing, tabulation, presentation,
analyzing and interpreting.
a) Descriptive – it deals with the data for purpose of describing
their characteristics. (i.e.) summarizing & presenting data.
b) Inferential – forecasts, estimates larger group of data from
sample data.
2. Applied Statistics – application of statistical rules & principles
to concrete factors like wages, income, population. (Quality
control, Sample surveys, etc.)

Objectives of Statistics:
The main objective of statistics are –
 To study the population and variables to make decisions and solve
problems.
 To make sense from population or mass.
 To take action on basis of available data.
 To bear light on complexity of problem.
 To forecast the future trend from data.
 To prove unknown from known data.
 To examine changes in particular activities.
 To draw conclusions from information.
 To provide basis for formation of knowledge relating to a particular
field of study.

Importance of Statistics:
The major importance of statistics are -
Statistics are the eyes of administration. (all business need
adequate data before judgements)
Statistics are aids to supervision. (it is tool for supervision of
work in obtaining efficiency of employees)
Statistics are invaluable in business. (estimates demand for
products in market, that is, help in planning and policy
making)
Statistical methods are indispensable in a quantitative study.
(useful in marketing, accounting and operating activities)

Functions of Statistics:
The following are the main functions of statistics –
 It simplifies complex mass of data in an intelligible manner.
 It enlarges individual experience that helps in making
decisions.
 It indicates tendencies or trends or positions or directions of
changes in data.
 It collects the data systematically in a definite form, as
information, useful for various purposes.
 It presents data in a most suitable manner that can be
understood at a glance.

Functions of Statistics:
 It compares one set of data with the other and discloses the
comparative position.
 It studies or establishes relationship between two related aspects
of particular phenomenon.
 It guides the management in formulating the plans and policies.
 It acts as a guide in measuring the effects of government
policies and business.
 It assists in testing the hypothesis in theory and discovering new
theories.
 It helps in estimating the present and forecasting future
activities.

Scope of Statistics:
The following points explain the scope of statistics –
1. Statistics and State:
Its objective was to collect data relating to population,
manpower, wealth, etc.
The concept of state indicates all welfare activities of
government departments like finance, transport, commerce,
defence, etc.
2. Statistics and Business:
It includes tools like central tendency, regression, time series,
etc. to provide accurate and timely information to managers
and used in all activities of business.

3. Statistics and Economics:
Statistical data and statistical methods are of great help in
proper understanding of economic problems in
formulation of economic policies.
Compilation of population data, calculation of income, per
– capita income, exports, imports, business cycle, etc. are
done through statistical methods.
4. Statistics and Natural Sciences:
Statistical techniques are used in natural sciences like
biology, medicine, zoology, astronomy, etc.

5. Statistics and Social Sciences:
It is the science of measurement of social organism and
reflects on importance of statistics in social sciences.
Statistical techniques are used in discipline of art,
psychology, education, etc.
6. Statistics and Research:
It is indispensable in research work and many statistical
techniques like chi square, ANOVA, correlation , T test are
used in analysis and interpretation of research findings.
It enable to solve many problems in almost all fields.

Limitations of Statistics:
The important limitations of statistics are –
1. Statistics deals only with quantitative data.
2. Statistics does not deal with individual facts.
3. Statistics laws are not exact.
4. Statistics tools do not provide the best solution
under all circumstances.
5. Statistics are liable to be misused.

Definition:
“Classification is the process of arranging the data into
sequences and groups according to their common characteristics
or separating them into different but related parts.”
CLASSIFICATION OF DATA
Classification of Data
Qualitative
(Attributes or Descriptive)
Quantitative
(Variables or Numerical)
Simple Composite Arbitrary
Discrete
(integers)
Continuous
(fractional)

1. Qualitative data – data classified according to characteristics or
qualities or some properties.
a) Simple – presence of single attributes. (E.g.) Gender
b) Composite – presence of more than one attributes. (E.g.) Gender
with Martial status or Education.
c) Arbitrary – not clearly defined and differ from person to person.
(E.g.) Tall or short persons, young and old.
2. Quantitative data – data classified according to quantitative
measurements like age, weight, prices, income, etc.
a) Discrete – it takes only integers, definite integer and no continuity.
b) Continuous – all possible values, integer, fractional and has
continuity.

Definition:
“Tabulation is a process of systematic and orderly
presentation of classified statistical data for a quick location of
desired information, in columns and rows.”
Parts of a Table:
* Table Number * Title * Captions
* Body
* Head note (preferably placed in brackets – in rupees, in 000’s)
* Footnote (using asterisk * to show explanation is given below.
* Source Note
TABULATION OF DATA

Exercise 1:
Draw a neat table to present data relating to no. of college
students according to faculty, semester & Gender.
Exercise 2:
Draw a table showing the no. of employees in Canara Bank
according to age and gender.
Exercise 3:
A super market divided into three sections grocery, vegetables &
novelty and sales in grocery in 2019 were Rs.50,000 & 5%
increase in 2020, vegetables in 2019 were Rs.75000 & 10%
increase in 2020 and novelty in 2019 were Rs.60000 & 8%
increase in 2020.

Frequency:
“Frequency is a number of times each value of variable
occurs in the series.”
It refers to the number of repetitions of a particular value of
variable.
Frequency Distribution:
“Frequency distribution is a summary presentation of
values of variables (or attributes) arranged according to their
magnitude either individually or in groups or in classes.
(i.e.) by counting frequencies.
FREQUENCY & FREQUENCY
DISTRIBUTION

Example:
Sam played football on Saturday morning, Saturday Afternoon
and Thursday Afternoon.
Solution:
The frequency of playing football was 2 times on Saturday, 1
time on Thursday and 3 for the whole week.
Frequency Table:
Day No. of times played
Saturday 2
Thursday 1
Total 3 (in a week)

A frequency distribution is constructed for the three main
reasons:
1. To facilitate the analysis of data.
2. To estimate frequencies of the unknown population
distribution from the distribution of sample data.
3. To facilitate the computation of various statistical measures.
Types of Frequency distributions:
1. Discrete (discontinuous) or Ungrouped frequency
distribution.
2. Continuous (Grouped) frequency distribution.

1. Discrete (discontinuous) or ungrouped frequency distribution:
 The variables which can take only definite or particular integers
are called ‘discrete’ frequency distribution.
 This is facilitated through technique called ‘Tally bars’ or ‘Tally
marks’.
Steps for constructing discrete frequency table:
1. In first column, all values of variables are placed in ascending
order.
2. In second column, tally bars are marked against each variable,
after occurring four times (4 tally bars), fifth tally bars should be
mentioned in cross line, which indicates the no. of occurrences.
3. In third column, total no. of tally bars entitled ‘Frequency’.

Exercise Problems (Discrete):
1. In a survey of 40 families in a village, the number of
children per family was recorded and the following data
obtained.
1 0 3 2 1 5 6 2
2 1 0 3 4 2 1 6
3 2 1 5 3 3 2 4
2 2 3 0 2 1 4 5
3 3 4 4 1 2 4 5

Solution:
Frequency distribution of the number of children are -

2. Continuous (Grouped) frequency distribution:
 When the number of observations and number of values of
variable both are large in size and consists of continuous variables
are called ‘continuous’ frequency distribution.
 The data condensed by dividing the entire range of value of
variables into suitable groups or class.
Class Limits:
“The class limits are the lowest and highest values of
variables that can be included in a class”.
* Lower Limit – it is the value below which there can be no item in
the class.
* Upper Limit – it is the value above which there can be no item in
the class.

Class Interval:
◦The class interval may be defined as the size of each
grouping of data. For example, 50-75, 75-100, 100-125…
are class intervals.

Width or size of the class interval:
◦ The difference between the lower and upper class limits is
called Width or size of class interval and is denoted by ‘ C’

Range:
◦The difference between largest and smallest value of the
observation is called The Range.
◦It is denoted by ‘ R’
 R = Largest value – Smallest value
 R = L - S

Mid-value or Middle Point:
◦The central point of a class interval is called the mid value or
mid-point.
◦It is found out by adding the upper and lower limits of a class
and dividing the sum by 2.
Mid-Value = L + U / 2
◦(E.g.) If the class interval is 20-30 then the mid- value is
20+30 / 2 = 25

Exercise Problems (Continuous):
1. The statistical data collected are generally raw data or
ungrouped data. Let us consider the daily wages (in Rs.) of 30
labourers in a factory.
80 70 55 50 60 65 40 30 80 90
75 45 35 65 70 80 82 55 65 80
60 55 38 65 75 85 90 65 45 75

Solution:
Arrangement of data in ascending order
30 35 38 40 45 45 50 55 55 55
60 60 65 65 65 65 65 65 70 70
75 75 75 80 80 80 80 85 90 90

Homework problems:
1. The following data gives the number of children in 50
families. Construct a discrete frequency table.
4 2 0 2 3 2 2 1 0 2
3 5 1 1 4 2 1 3 4 2
6 1 2 2 2 1 3 4 1 0
1 3 4 1 0 1 2 2 2 5
2 4 3 0 1 3 6 1 0 1

Homework problems:
2. Thirty AA batteries were tested to determine how long they
would last. The results, to the nearest minute, were recorded as
follows:
Construct a frequency distribution table.
423 369 387 411 393 394 371 377
389 409 392 408 431 401 363 391
405 382 400 381 399 415 428 422
396 372 410 419 386 390 - -

Definition:
“Diagrams are visual aids which comprise of presenting
statistical materials in pictures, geometric figures and curves.”
Utilities of Diagrams:
Give more attractive presentation of data given by figures.
Create more stable effects on minds of the readers.
Simplify complex data and present the information
attractively.
They save time and drawing inferences from figures.
DIAGRAMS

Limitations of Diagrams:
 Diagrams have certain limitations –
 Utility to common man & utility to expert is limited.
 Limited size of information & fail to furnish detailed
information.
 Disclose only approximate values & don’t give accurate facts &
figures.
 Present data only in particular range.
 Taken into account only two or three sets of data.
 They are not subject to further mathematical analysis.
 They are not reliable sources and only means to draw
conclusions.

Types of Diagrams:
The following are the common methods of diagrams –
1. On the basis of Dimension:
a) One – Dimensional diagrams (lines and bars)
b) Two – Dimensional diagrams (squares and rectangles)
c) Three – Dimensional diagrams (cubes, cylinders and
blocks)
2. On the basis of View:
a) Pictograms
b) Cartograms (Mapograms)

1. On the basis of Dimension:
a) One – Dimensional diagrams (lines and bars)
Only one dimension of the figure is taken into account. Bars
with different widths and lengths.
i) Line Diagrams:
These diagrams are used when there is a large number of
values of variables with variations in their values within a
small range.
They are in form of vertical lines relating to respective
values of variable.

ii) Simple Bar Diagrams:
 These diagrams can be drawn either vertically or horizontally.
 Bar must have similar width and uniform space between two
bars.
 Values of variable are taken in either ascending or descending
order.
55
68
60
40
0
10
20
30
40
50
60
70
80
Karnataka TN Kerala MP
No.
of
SSI
Name of the State
No. of SSI
Karnataka
TN
Kerala
MP

iii) Multiple Bar Diagrams:
 These diagrams are also known as ‘compound bar diagrams.
 These diagrams adopted when two or more phenomena over a
no. of years are compared with each other.
 Different colours or shades or dots are used for each attribute.
88%
90%
95%
85%
92%
95%
80%
82%
84%
86%
88%
90%
92%
94%
96%
2018 2019 2020
Overall
percentage
Year
Overall Result Analysis
B.Com
BBA

iv) Sub - divided Bar Diagrams:
 These diagrams are also known as ‘component bar diagrams.
 Each bar is sub – divided according to components consisting in
it.
 Complete bar represents total values of variables along with
various values of components.
400
150 80 40
600
200
100
60
0
200
400
600
800
1000
1200
Income Food Clothing Rent
Family Budgets
Family A Family B

v) Sub - divided Percentage Bar Diagrams:
 In this diagrams, it converts the values of variables into
percentages.
 Now all bars look equal in heights representing the value of 100
as a percentage.
45% 35% 20%
44% 34%
22%
48% 36% 16%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Men Women Children
2020
2019
2018

vi) Deviation Bar Diagrams:
 These bars depict the net deviations in different values.
 The positive deviations are taken above the axis and negative
deviations taken below it.
 These diagrams are also known as “Bilateral Bar Diagrams”.

vii) Duo - Directional Bar Diagrams:
 These diagrams are drawn to show an aggregate result of
different and opposite components of the same phenomenon.
 The total value of variable is separated into two parts so that one
part lies above the axis and the other below the axis.

viii) Paired Bar Diagrams:
 These diagrams are drawn to show related data of same
phenomenon.
 The paired bars relate to each other and they are jointly studied.
 The bars are separated by axis vertically whereas paired bars are
separated horizontal by items to which data are related.

b) Two – Dimensional Diagrams:
In two – dimensional diagrams, length and breadth are taken
into consideration in drawing diagram to represent data.
These diagrams are also called “Surface Diagrams” or
“Area Diagrams”.
i) Rectangles:
A rectangle is a four sided figure with four right angles with
adjacent sides unequal.
It represents the relative magnitudes of two or more values.
They are placed side by side like bars and are modified form
of bar diagrams.

Rectangle Diagrams:
0
10
20
30
40
50
60
70
Karnataka
TN
Kerala
MP

ii) Squares:
Squares are figures with four equal sides and four right
angles.
Values of variable bear the long range of ratios like 1 : 100
or 4 : 400, square diagrams are applied.
Two comparable values of
variable can be represented by
square root of area of one side of
square.

iii) Circles:
 A Circle refers to the space enclosed by a curved line which
keeps the same distance from the Centre.
 The area of circle is proportional to the square of its radius.
 The circle diagrams are also called ‘Circular diagrams’.
Simple Circle diagrams:
 Area of circle varies as the square of its radius.
 Areas of the circles would also be in the same proportion as the
areas of the squares.
Sub divided Circle diagrams: (Angular or Pie charts)
 It is divided into different segments of circle based on different
attributes of data.

Simple Circle Diagram Sub divided circle diagram

c) Three – Dimensional Diagrams:
Three dimensional diagrams are cone, cubes, cylinder,
blocks, etc.
0
10
20
30
40
50
60
70
No.
of
SSI
State
No of SSI
Karnataka
TN
Kerala
MP
0
10
20
30
40
50
60
70
Karnataka
TN
Kerala
MP

1. On the basis of View:
a) Pictograms:
“Pictogram” is a device of picture by which data can be
presented.
This is called ‘Vienna Method’ or
‘Isotype method’.
It is used for comparing statistical
data.

b) Cartograms:
The different types of maps are used to present the data
instead of picture.
Data are shown in different
colours, shades, points or dots
having different attributes.
It is like Atlas map, depicting
the data relating to the
various parts of the world.

Definition:
“Graph is a vivid or intense or bright form of presentation
of data. It is a simplest and commonest aid to numerical reading
which gives a picture of numbers in such a way that the
relations between two series can be easily compared.”
Utilities of Diagrams:
It depicts the data more attractive than a table.
It depicts comparison of two or more series.
Well designed graphs are more effective in creating interest in
minds of readers.
It brings out hidden facts and relationships existing in data.
GRAPHS

Difference between Diagrams and Graphs:
Diagrams Graphs
Diagrams can be drawn on plane
papers & graph papers
Graphs can only be drawn on
graph papers
Diagrams, lines, rectangles,
circles, cubes and maps are used
Graphs, dots, dashes, curves are
used
Diagrams furnish approximate
information
Graph furnish more accurate
information
It depicts categorical &
geographical data
It depict time series and
frequency distribution
It requires some drawing skill Graphs can be drawn easily

Construction of Graph:
 A Graph sheet is a paper in which lines are drawn dividing
every inch or centimeter into 10 equal parts.
 A set of intersecting lines are also drawn at right angles.
 The horizontal lines are used ‘X’ – axis (abscissa) and
vertical lines are used ‘Y’ – axis (ordinates).
 These two axes divide the region of the plane into four parts
which are called ‘Quadrants’.
 Most of the statistical data are represented in the quadrant I
and IV.

Types of Graphs:
Graphs are generally classified into two categories –
1. Graphs of Time series:
a) One variable graph
b) Two variable graph
c) Three variable graph
2. Graphs of Frequency Distribution:
a) Histogram
b) Frequency polygon
c) Frequency Curve d) Ogive Curves

1. Graphs of Time Series:
Time series or historical series stands for the numerical
record of the changes in a variable during a given period of
time.
Time units are placed on X axis and values of variables on
Y axis.
All the points are connected by a continued smoothed lines
are called ‘Curve’.
All the points are connected by straight lines as an
alternative methods to a curve.

a) One Variable graph:
 Only one factor is shown on the Y axis and the time is
measured on X axis.
0
50
100
150
200
250
300
350
1991 - 1992 1992 - 1993 1993 - 1994 1994 - 1995
Rice Production
Rice Production

b) Two Variable graph:
 Two factor is shown on the Y axis and the time is measured
on X axis.
3300
4000
5700
6300
2000
2500
2800
3000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
1991 - 1992 1992 - 1993 1993 - 1994 1994 - 1995
Imports & Exports of India
Imports
Exports

c) Three Variable graph:
 Three factor is shown on the Y axis and the time is
measured on X axis.
150
180
160
190
90
100 120
190
60
80
40
0
0
50
100
150
200
250
300
350
400
1 2 3 4
Expenditure & Balance
Profit / Loss
Expenditure
Income

2. Graphs of Frequency Distribution:
When data is expressed in terms of occurrence of
frequencies, it is essential to draw a frequency graph.
The values of variables or mid – values of class interval on
X axis and frequencies on Y axis.
a) Histogram:
Histogram is a device of graphic representation of a
frequency distribution.
It is constructed by erecting set of rectangles on each class
interval and on horizontal respective class frequencies.
It is also called ‘Staircase or Block Diagram’.

Histogram:
0
5
10
15
20
25
30
35
35-40 40-45 45-50 50-55
Axis
Title
Axis Title
Frequency

b) Frequency Polygon:
 It is a device of graphic presentation of a frequency
distribution.
 It is a simple method of drawing the graph with the help of
histogram.
 First construct the histogram and plot the mid points of top
of each rectangle.
 To make frequency polygon, connect the mid point of top
of all rectangles by straight line.
 Area of frequency polygon is equal to area of histogram.

c) Frequency Curve:
 With the help of histogram and frequency polygon, we can
draw smoothed curve to eliminate the irregularities in data.
 A smoothed frequency curves represents a generalized
characterization of data collected from population or mass.

d) Ogive Curves:
 These curves refer to a continuous form of cumulative
frequency curves less than cumulative frequency curve and
more than cumulative frequency curve.
 This method of drawing the curves is best among other
types and its called ‘Cumulative Frequency Curves’.
 Ogive curve shows rising trend (less than frequency) or
falling trend (more than frequency).
 Ogive curves used for purpose of comparing groups of
statistics in which time is not a factor.

Types of Ogive Curves:
i) Less than Ogive – it consists in plotting the ‘less than’
frequencies against upper limit of class interval or
boundaries.
It is increasing curve sloping upward from left to right
of graph and it is in shape of elongated (S).
ii) More than Ogive – it consists in plotting the ‘more than’
frequencies against lower limit of class interval or
boundaries.
It is decreasing curve sloping downward from left to
right of graph and it is in shape of elongated upside down.

Introduction:
Probability theory is concerned with the study of
random (or chance) phenomena, such phenomena
are characterized by the fact that their future
behavior is nor predictable in a deterministic
fashion.
Probability is a numerical measure of the likehood
of an occurrence of event. It is a measure of the
degree of uncertainty associated with random
events.
PROBABILITY

Basic Concepts of Probability:
Random Experiment:
 It is an experiment which can be repeated any number of
items under the same conditions, but does not give unique
results. The result will be any one of several possible
outcomes, but for each trial, the result will not be known in
advance. A random experiment is also called a trail and the
outcomes are called events.
(E.g.) Rolling a dice is a trial, getting 2 is an event.
Tossing a coin is a trial, getting head is an event.

Sample Space:
 The total of all possible outcomes of a random experiment is called
a sample space (S) and a possible outcome, or element in a sample
space is called a Sample Point.
(E.g.) In throwing a dice, S = {1, 2, 3, 4, 5, 6}
Exhaustive events:
 All possible outcomes of an experiment are called exhaustive
events.
Favourable events:
 The number of cases favourable to an event in a trial is the number
of outcomes which entail the happening of the event. (E.g.) In
drawing a card from a deck of cards, the number of favourable
cases in getting a spade is 13.

Equally likely events:
• Two or more events are equally likely, if each of them has
an equal chance of happening.
Mutually exclusive events:
• Two events are said to be mutually exclusive if the
occurrences of any one of them excludes the occurrence of
the other in a single experiment.
• (E.g.) If a coin is tossed, the events Head (H) and Tail (T)
are mutually exclusive.

Independent events:
• Two or more events are independent, if the occurrence of one does
not affect the occurrence of the other.
• (E.g.) If a coin is thrown twice, the result of the second throw is not
affected by the result of the first throw.
Dependent events:
• Two events are said to be dependent if the occurrence or non –
occurrence of an event in a trail affects the occurrence of the other
event in other trails.
• (E.g.) If we draw 2 cards one after the other a pack, we draw one card
out of 52 cards in the first case. In the second case we draw one card
out of 51 cards. Thus the two events are dependent. But if the first
card is replaced before the second draw, the events are independent.

Complementary events:
• If A and B are mutually exclusive and exhaustive events,
then A is the complementary event of B and vice versa.
• (E.g.) When a dice is thrown, occurrence of an even number
and occurrence of an odd number are complementary events.
Definition of Probability (Mathematical)
• If there are ‘m’ equally likely, mutually exclusive and
exhaustive outcomes and ‘m’ of them are favourable to an
event A, then the probability of the happening of A is

Definition of Probability (Statistical)
If an experiment is repeated a large number of times under
essentially identical and homogeneous conditions, then the limiting value
of the ratio of the number of times the event A happens to the total
number of trails of the experiments, as the number of trails increases
indefinitely is called the probability of the occurrence of A.
• If the event A happens ‘m’ times out of ‘n’ repetitions of a random
experiment, then

Probability – Exercise Problems
1. Three coins are tossed together. Find the probability
that there are exactly 2 heads.
2. What is the probability of getting a sum of 7 when
two dice are thrown?
3. Find the probability of getting a numbered card
when a card is drawn from the pack of 52 cards.
4. A bag contains 4 red, 5 white and 6 black balls.
What is the probability that two balls drawn are red
and black.

Probability – Classwork Problems
1. 4 cards are drawn from a well shuffled pack of
cards. Find the probability that
i) all the four queens
ii) there is one card from each unit
iii) two cards are diamonds and two are spades and
iv) all the four cards are heaters and one of them is a
jack.
2. From a pack of cards, one card is drawn. What is
the probability that it is either spade or a king?

Probability – Homework Problems
1. What is the probability of choosing a heart from a deck of
cards?
2. What is the probability of choosing a three from a deck of
cards?
3. Out of numbers 1 to 120, one is selected at random. What is
the probability that it is divisible by 8 or 10?
4. A bag contains 7 green, 4 white and 5 red balls. If four balls
are drawn one by one without replacement. What is the
probability that none is red?
5. 4 persons are chosen at random from a group containing 3
men, 2 women & 4 children. What is the probability of getting
exactly two of them are children?

Conditional Probability and
Independence

Conditional Probability – Exercise Problems

Conditional Probability – Exercise Problems
3. A is known to hit the target in 2 out of 5 shots, whereas
B is known to hit the target in 3 out of 4 shots. Find the
probability of the target being hit when they both try.
4. An article manufactured by a company consists of two
parts A and B. In the process of manufacture of part A, 9
out of loo are likely to be defective. Similarly 5 out of 100
are likely to be defective in the manufacture of B.
Calculate the probability that the assembled part will not
be defective.

Introduction to Business Statistics

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Introduction to Business Statistics