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Biju Patnaik University of Technology BPUT
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MCA
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**Master of Computer Applications**Offline Downloads:
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MCA
SEMESTER - II
PROBABILITY &
STATISTICS

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1
PROBABILITY
INTRODUCTION TO PROBABILITY
Managers need to cope with uncertainty in many decision
making situations. For example, you as a manager may assume
that the volume of sales in the successive year is known exactly to
you. This is not true because you know roughly what the next year
sales will be. But you cannot give the exact number. There is some
uncertainty. Concepts of probability will help you to measure
uncertainty and perform associated analyses. This unit provides the
conceptual framework of probability and the various probability
rules that are essential in business decisions.
Learning objectives:
After reading this unit, you will be able to:
Appreciate the use of probability in decision making
Explain the types of probability
Define and use the various rules of probability depending on
the problem situation.
Make use of the expected values for decision-making.
Probability
Sets and Subsets
The lesson introduces the important topic of sets, a simple
idea that recurs throughout the study of probability and statistics.
Set Definitions
A set is a well-defined collection of objects.
Each object in a set is called an element of the set.
Two sets are equal if they have exactly the same elements
in them.
A set that contains no elements is called a null set or an
empty set.
If every element in Set A is also in Set B, then Set A is a
subset of Set B.
Set Notation
A set is usually denoted by a capital letter, such as A, B, or
C.
An element of a set is usually denoted by a small letter, such
as x, y, or z.
A set may be decribed by listing all of its elements enclosed
in braces. For example, if Set A consists of the numbers 2,
4, 6, and 8, we may say: A = {2, 4, 6, 8}.
The null set is denoted by {∅}.

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Sets may also be described by stating a rule. We could
describe Set A from the previous example by stating: Set A
consists of all the even single-digit positive integers.
Set Operations
Suppose we have four sets - W, X, Y, and Z. Let these sets be
defined as follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3,
4}.
The union of two sets is the set of elements that belong to
one or both of the two sets. Thus, set Z is the union of sets X
and Y.
Symbolically, the union of X and Y is denoted by X ∪ Y.
The intersection of two sets is the set of elements that are
common to both sets. Thus, set W is the intersection of sets
X and Y.
Symbolically, the intersection of X and Y is denoted by X ∩
Y.
Sample Problems
1. Describe the set of vowels.
If A is the set of vowels, then A could be described as A = {a,
e, i, o, u}.
2. Describe the set of positive integers.
Since it would be impossible to list all of the positive
integers, we need to use a rule to describe this set. We
might say A consists of all integers greater than zero.
3. Set A = {1, 2, 3} and Set B = {3, 2, 1}. Is Set A equal to Set
B?
Yes. Two sets are equal if they have the same elements.
The order in which the elements are listed does not matter.
4. What is the set of men with four arms?
Since all men have two arms at most, the set of men with
four arms contains no elements. It is the null set (or empty
set).
5. Set A = {1, 2, 3} and Set B = {1, 2, 4, 5, 6}. Is Set A a subset
of Set B?
Set A would be a subset of Set B if every element from Set A
were also in Set B. However, this is not the case. The
number 3 is in Set A, but not in Set B. Therefore, Set A is not
a subset of Set B.
Statistical Experiments
All statistical experiments have three things in common:
The experiment can have more than one possible outcome.
Each possible outcome can be specified in advance.

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The outcome of the experiment depends on chance.
A coin toss has all the attributes of a statistical experiment. There is
more than one possible outcome. We can specify each possible
outcome (i.e., heads or tails) in advance. And there is an element of
chance, since the outcome is uncertain.
The Sample Space
A sample space is a set of elements that represents all
possible outcomes of a statistical experiment.
A sample point is an element of a sample space.
An event is a subset of a sample space - one or more
sample points.
Types of events
Two events are mutually exclusive if they have no sample
points in common.
Two events are independent when the occurrence of one
does not affect the probability of the occurrence of the other.
Sample Problems
1. Suppose I roll a die. Is that a statistical experiment?
Yes. Like a coin toss, rolling dice is a statistical experiment.
There is more than one possible outcome. We can specify each
possible outcome in advance. And there is an element of
chance.
2. When you roll a single die, what is the sample space.
The sample space is all of the possible outcomes - an integer
between 1 and 6.
3. Which of the following are sample points when you roll a die - 3,
6,
and
9?
The numbers 3 and 6 are sample points, because they are in
the sample space. The number 9 is not a sample point, since it
is outside the sample space; with one die, the largest number
that you can roll is 6.
4. Which of the following sets represent an event when you roll a
die?
A.
B.
C.
D.
{1}
{2, 4,}
{2, 4, 6}
All of the above
The correct answer is D. Remember that an event is a subset of
a sample space. The sample space is any integer from 1 to 6.

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