,
, The science of probability theory was first noticed in the middle of the 18th century with the study of
games of chance. As time went by, the applications of this science expanded to the point where today, the theory
of probability plays a crucial role in recognizing, modeling, and improving many uncertain real-world
phenomena.
The book in front of you is the result of several years of studying, using, and teaching the concepts of
probability theory and its applications. Despite the many complexities of this science, the main goal and focus
of this book are to present its contents simply and fluently and to help students create a deep understanding of
the subjects of this science. To achieve this goal, each chapter of the book is divided into two main parts. In the
first part, the text of each chapter is explained in a simple way and has examples that increase learning and
understanding of the content. Since the authors believe solving different problems is the best way to get good at
probability theory and fully understand it, the second half of each chapter in the book is filled with many
classified problems. The order of the problems is also designed such that the readers feel themselves progressing
along the learning path step by step.
In the first chapter of this book, the main ideas of combinatorial analysis, which is the primary way to
figure out probabilities, are explained.
In Chapter 2, the definition of probability, the principles of probability theory, and the main methods of
probability calculation are stated, and in Chapter 3, conditional probability and its applications are presented.
In Chapter 4, random variables are addressed, and in Chapter 5, the expected value of random variables
and some of their properties are mentioned. Also, in Chapters 6 and 7, some widely used discrete and continuous
random variables are introduced.
In Chapter 8, joint random variables and conditional distributions are discussed, and in Chapters 9 and
10, the distribution and expected value of a function of several random variables are addressed.
Finally, in Chapter 11, expected value of multiple random variables, correlation, and covariance are
explained.
For those who encounter difficulties in solving the problems in this book, another book titled "Solution
Manual for Principles of Probability and its Applications" has been prepared by the authors, in which there are
explanatory answers to all questions and, in many cases, more than one solution to the problems is proposed.
The contents of this book are not without flaws, and we incentivize our dear readers to contact us via
if they find any weak points or faults in the book or have any comments or
suggestions.
Ultimately, the authors hope that this collection can be effective in getting to know and learning the
science of probability better and be a step, however small, in developing this science.
, CHAPTER 1: COMBINATORIAL ANALYSIS ............................................................................................................................................... 1
1.1. Introduction ....................................................................................................................................................................................................... 1
1.2. The Basic Principle of Counting ................................................................................................................................................................... 1
1.3. Permutations .................................................................................................................................................................................................... 5
1.3.1 Permutation of “n” Distinct elements .......................................................................................................................................... 5
1.3.2 Permutation of “r” Distinct elements from “n” Distinct elements ........................................................................................ 7
1.3.3 Permutation of “n” Elements, some of which have the Same value...................................................................................... 8
1.3.4 Permutation of “n” Distinct elements at a Round table......................................................................................................... 10
1.4. Combinations ................................................................................................................................................................................................. 12
1.5. Significant Identities of the Combinatorial Topic ................................................................................................................................ 16
1.6. The Ball and Urn (cell) Model ..................................................................................................................................................................... 23
1.7. Chapter Problems ..........................................................................................................................................................................................48
CHAPTER 2: AXIOMS OF PROBABILITY ................................................................................................................................................ 58
2.1. Introduction .................................................................................................................................................................................................... 58
2.2. Random Trial, Sample Space, and Event ................................................................................................................................................ 58
2.3. An Introduction to the Algebra of Sets ..................................................................................................................................................... 59
2.4. Definition of Probability .............................................................................................................................................................................. 64
2.5. Some Probabilistic Propositions Resulting from Principles of the Probability Theory ............................................................... 79
2.6. Chapter Problems ..........................................................................................................................................................................................96
CHAPTER 3: CONDITIONAL PROBABILITY AND INDEPENDENCE ..................................................................................................... 106
3.1. Introduction .................................................................................................................................................................................................. 106
3.2. Conditional Probability Concept ............................................................................................................................................................. 106
3.3. The Law of Multiplication in Probability ............................................................................................................................................... 113
3.4. Independence of Events .............................................................................................................................................................................. 115
3.5. The Law of Total Probability .................................................................................................................................................................... 125
3.6. Bayes' Law ..................................................................................................................................................................................................... 133
3.7. The Law of Total Probability in Reduced Space .................................................................................................................................. 138
3.8. Chapter Problems ........................................................................................................................................................................................ 143
CHAPTER 4: RANDOM VARIABLES ...................................................................................................................................................... 162
4.1. Introduction .................................................................................................................................................................................................. 162
4.2. Types of Random Variables....................................................................................................................................................................... 166
4.3. Discrete Random Variables ....................................................................................................................................................................... 167
4.4. Continuous Random Variables ................................................................................................................................................................ 169
4.5. Mixed Random Variables ...........................................................................................................................................................................174
4.6. Cumulative Distribution Function ......................................................................................................................................................... 176
4.7. Some Important Values of Random Variables ...................................................................................................................................... 181
4.8. The Distribution of a Function of a Random Variable ....................................................................................................................... 184
4.9. Conditioning on Continuous Space ........................................................................................................................................................ 188
4.10. Chapter Problems ......................................................................................................................................................................................... 191
CHAPTER 5: EXPECTED VALUE .......................................................................................................................................................... 206
5.1. Introduction ................................................................................................................................................................................................. 206
5.2. The Expected Value of Discrete, Continuous, and Mixed Random Variables ............................................................................. 209
I
, The science of probability theory was first noticed in the middle of the 18th century with the study of
games of chance. As time went by, the applications of this science expanded to the point where today, the theory
of probability plays a crucial role in recognizing, modeling, and improving many uncertain real-world
phenomena.
The book in front of you is the result of several years of studying, using, and teaching the concepts of
probability theory and its applications. Despite the many complexities of this science, the main goal and focus
of this book are to present its contents simply and fluently and to help students create a deep understanding of
the subjects of this science. To achieve this goal, each chapter of the book is divided into two main parts. In the
first part, the text of each chapter is explained in a simple way and has examples that increase learning and
understanding of the content. Since the authors believe solving different problems is the best way to get good at
probability theory and fully understand it, the second half of each chapter in the book is filled with many
classified problems. The order of the problems is also designed such that the readers feel themselves progressing
along the learning path step by step.
In the first chapter of this book, the main ideas of combinatorial analysis, which is the primary way to
figure out probabilities, are explained.
In Chapter 2, the definition of probability, the principles of probability theory, and the main methods of
probability calculation are stated, and in Chapter 3, conditional probability and its applications are presented.
In Chapter 4, random variables are addressed, and in Chapter 5, the expected value of random variables
and some of their properties are mentioned. Also, in Chapters 6 and 7, some widely used discrete and continuous
random variables are introduced.
In Chapter 8, joint random variables and conditional distributions are discussed, and in Chapters 9 and
10, the distribution and expected value of a function of several random variables are addressed.
Finally, in Chapter 11, expected value of multiple random variables, correlation, and covariance are
explained.
For those who encounter difficulties in solving the problems in this book, another book titled "Solution
Manual for Principles of Probability and its Applications" has been prepared by the authors, in which there are
explanatory answers to all questions and, in many cases, more than one solution to the problems is proposed.
The contents of this book are not without flaws, and we incentivize our dear readers to contact us via
if they find any weak points or faults in the book or have any comments or
suggestions.
Ultimately, the authors hope that this collection can be effective in getting to know and learning the
science of probability better and be a step, however small, in developing this science.
, CHAPTER 1: COMBINATORIAL ANALYSIS ............................................................................................................................................... 1
1.1. Introduction ....................................................................................................................................................................................................... 1
1.2. The Basic Principle of Counting ................................................................................................................................................................... 1
1.3. Permutations .................................................................................................................................................................................................... 5
1.3.1 Permutation of “n” Distinct elements .......................................................................................................................................... 5
1.3.2 Permutation of “r” Distinct elements from “n” Distinct elements ........................................................................................ 7
1.3.3 Permutation of “n” Elements, some of which have the Same value...................................................................................... 8
1.3.4 Permutation of “n” Distinct elements at a Round table......................................................................................................... 10
1.4. Combinations ................................................................................................................................................................................................. 12
1.5. Significant Identities of the Combinatorial Topic ................................................................................................................................ 16
1.6. The Ball and Urn (cell) Model ..................................................................................................................................................................... 23
1.7. Chapter Problems ..........................................................................................................................................................................................48
CHAPTER 2: AXIOMS OF PROBABILITY ................................................................................................................................................ 58
2.1. Introduction .................................................................................................................................................................................................... 58
2.2. Random Trial, Sample Space, and Event ................................................................................................................................................ 58
2.3. An Introduction to the Algebra of Sets ..................................................................................................................................................... 59
2.4. Definition of Probability .............................................................................................................................................................................. 64
2.5. Some Probabilistic Propositions Resulting from Principles of the Probability Theory ............................................................... 79
2.6. Chapter Problems ..........................................................................................................................................................................................96
CHAPTER 3: CONDITIONAL PROBABILITY AND INDEPENDENCE ..................................................................................................... 106
3.1. Introduction .................................................................................................................................................................................................. 106
3.2. Conditional Probability Concept ............................................................................................................................................................. 106
3.3. The Law of Multiplication in Probability ............................................................................................................................................... 113
3.4. Independence of Events .............................................................................................................................................................................. 115
3.5. The Law of Total Probability .................................................................................................................................................................... 125
3.6. Bayes' Law ..................................................................................................................................................................................................... 133
3.7. The Law of Total Probability in Reduced Space .................................................................................................................................. 138
3.8. Chapter Problems ........................................................................................................................................................................................ 143
CHAPTER 4: RANDOM VARIABLES ...................................................................................................................................................... 162
4.1. Introduction .................................................................................................................................................................................................. 162
4.2. Types of Random Variables....................................................................................................................................................................... 166
4.3. Discrete Random Variables ....................................................................................................................................................................... 167
4.4. Continuous Random Variables ................................................................................................................................................................ 169
4.5. Mixed Random Variables ...........................................................................................................................................................................174
4.6. Cumulative Distribution Function ......................................................................................................................................................... 176
4.7. Some Important Values of Random Variables ...................................................................................................................................... 181
4.8. The Distribution of a Function of a Random Variable ....................................................................................................................... 184
4.9. Conditioning on Continuous Space ........................................................................................................................................................ 188
4.10. Chapter Problems ......................................................................................................................................................................................... 191
CHAPTER 5: EXPECTED VALUE .......................................................................................................................................................... 206
5.1. Introduction ................................................................................................................................................................................................. 206
5.2. The Expected Value of Discrete, Continuous, and Mixed Random Variables ............................................................................. 209
I
