PROBABILITY I
Department of Statistics
Ranaghat College
,Contents
1 Introduction 2
1.1 Random Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Trial and Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Equally Likely Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Exhaustive Cases or Total Number of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Favourable Number of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Classical or Mathematical Definition of Probability 4
2.1 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Limitations of Classical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Statistical regularity 7
3.1 Limitations of Statistical Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Sets and Set Operation 8
4.1 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Compliment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Disjoint Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.6 A 4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.7 The Operations of Union and Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.8 Classes of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Axiomatic Definition of Probability 13
6 Conditional Probability 18
7 Bayes’ Theorem 22
8 Independence of Events 25
8.1 Pairwise Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8.2 Mutually Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1
, Chapter 1
Introduction
1.1 Random Experiment
The word experiment is used to describe an act which can be repeated under some given essentially homo-
geneous condition (same condition).
An experiment perform without the definite prediction about the result is known as random experiment.
Example: Whenever we toss a coin or throw a die we cannot predict whether it will be head or tail or which
face of the die will come up to the top. So these are all the examples of random experiment.
1.2 Trial and Event
If an experiment be perform under essentially the same physical condition then each experiment is called
a trial and outcome and combinations of outcomes then obtained are termed as events or cases. Thus the
outcomes of a random experiment are known as random events. Example: Drawing of two balls from an
urn containing ‘a’ red and ‘b’ black balls is a trial and getting of both red balls or both black balls or one red
and one black are events.
The events may be elementary or composite. Elementary events are those which are single in nature. In
case of throwing a die, face 5 or face 3 etc. are the elementary events. Composite events are those which are
formed through the combination of two or more elementary events. In case of throwing a die the event ‘odd
number of faces’ is a composite event as it is the combination of face 1, face 3, face 5.
Events are denoted by capital letters say A, B, C etc. or X, Y, Z etc.
1.3 Mutually Exclusive Events
Events are said to be mutually exclusive if two or more of them cannot occur simultaneously. Example: In
throwing a die all the 6 faces numbered 1, 2, 3, 4, 5, 6 are mutually exclusive.
1.4 Equally Likely Cases
The outcomes of a random experiment are said to be equally likely, if after taking into considerations of all
relevant evidence, none of them is expected to occur in preference to other. Example: If a die is unbiased
there is no reason to expect that in any throw some particular face will come up more frequently than other
faces.
2
Department of Statistics
Ranaghat College
,Contents
1 Introduction 2
1.1 Random Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Trial and Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Equally Likely Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Exhaustive Cases or Total Number of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Favourable Number of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Classical or Mathematical Definition of Probability 4
2.1 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Limitations of Classical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Statistical regularity 7
3.1 Limitations of Statistical Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Sets and Set Operation 8
4.1 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Compliment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Disjoint Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.6 A 4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.7 The Operations of Union and Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.8 Classes of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Axiomatic Definition of Probability 13
6 Conditional Probability 18
7 Bayes’ Theorem 22
8 Independence of Events 25
8.1 Pairwise Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8.2 Mutually Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1
, Chapter 1
Introduction
1.1 Random Experiment
The word experiment is used to describe an act which can be repeated under some given essentially homo-
geneous condition (same condition).
An experiment perform without the definite prediction about the result is known as random experiment.
Example: Whenever we toss a coin or throw a die we cannot predict whether it will be head or tail or which
face of the die will come up to the top. So these are all the examples of random experiment.
1.2 Trial and Event
If an experiment be perform under essentially the same physical condition then each experiment is called
a trial and outcome and combinations of outcomes then obtained are termed as events or cases. Thus the
outcomes of a random experiment are known as random events. Example: Drawing of two balls from an
urn containing ‘a’ red and ‘b’ black balls is a trial and getting of both red balls or both black balls or one red
and one black are events.
The events may be elementary or composite. Elementary events are those which are single in nature. In
case of throwing a die, face 5 or face 3 etc. are the elementary events. Composite events are those which are
formed through the combination of two or more elementary events. In case of throwing a die the event ‘odd
number of faces’ is a composite event as it is the combination of face 1, face 3, face 5.
Events are denoted by capital letters say A, B, C etc. or X, Y, Z etc.
1.3 Mutually Exclusive Events
Events are said to be mutually exclusive if two or more of them cannot occur simultaneously. Example: In
throwing a die all the 6 faces numbered 1, 2, 3, 4, 5, 6 are mutually exclusive.
1.4 Equally Likely Cases
The outcomes of a random experiment are said to be equally likely, if after taking into considerations of all
relevant evidence, none of them is expected to occur in preference to other. Example: If a die is unbiased
there is no reason to expect that in any throw some particular face will come up more frequently than other
faces.
2
