Probability is extremely important in today's society, and the theory has several practical applications. For example, scientists utilize the notion of probability to predict the risk of a pandemic emerging in the future, and doctors use probability theory to estimate the likelihood that a patient would react to a specific therapy or die from the sickness. In reality, it is no longer news that many individuals make a lot of money by gambling and betting on games, and probability, which started in gambling, has become a highly valuable instrument that gamblers use to stack the odds in their favor and increase their gambling revenue. It is because probability has become so valuable in our society and is extensively taught in high schools, colleges, and universities, that I choose to focus this book on theory and practice of probability.
Probabilitatem Extraordinaire is a textbook that I wrote based on my 15 years of teaching experience. In my teaching experience, I observed that some students study at a very slow rate, some at a fast pace, and some at an average pace. Furthermore, nearly all students who learn at slow or medium speeds are numerophobic, which means they have a strong phobia or dread of studying mathematics and other calculative courses, and this fear is known as arithmophobia. As a result, Probabilitatem Extraordinaire was composed with the understanding that individuals have varying learning speeds, particularly those who learn slowly.
The text was carefully compiled to ensure that even students with no prior exposure to math-related subjects or those apprehensive about STEM-related subjects can comprehend the entire content of this book should they choose to engage with it. An intriguing aspect of this book is that not only is it presented in a highly simplified manner, but it is also designed to be exceptionally appealing to readers. Consequently, any learner who encounters this book for the first time will be enticed to read it in its entirety.
Hence, Probabilitatem Extraordinaire was created with the aim of assisting every student in overcoming their fear of arithmetic. It caters to quick learners by incorporating highly advanced probability concepts, exposing all students to high-level ideas that will help them excel in settings requiring advanced prediction and forecasting skills. This book will be highly beneficial for high school math teachers and department lecturers focusing on probability. It contains practical examples.
Table of Contents
CHAPTER ONE
1.1: THEORY OF PROBABILITY: THE DOCTRINE OF CHANCES
1.2: TERMINOLOGIES USED IN PROBABILITY
1.2.1: EXPERIMENT AND TRIAL
1.2.2: RANDOM EXPERIMENT
1.2.3: OUTCOME
1.2.4: EVENT (E)
1.2.5: SAMPLE SPACE (S)
1.3: PRACTICAL ILLUSTRATION OF THESE TERMINOLOGIES
1.6: PRACTICAL EXAMPLE ONE
1.7: RANDOM VARIABLE X
1.7.1: DISCRETE AND CONTINUOUS RANDOM VARIABLE
1.8: EXPECTED VALUES --- A BRIEF INTRODUCTION
1.8.1: EXPECTATION OF A DISCRETE RANDOM VARIABLE
1.9: PRACTICAL EXAMPLE TWO
1.10: PRACTICAL EXAMPLE THREE
1.11: EXPECTED VALUE OF CONTINUOUS RANDOM VARIABLE
1.12: PRACTICAL EXAMPLE FOUR
1.13: PROPERTIES OF EXPECTATION
1.14: VARIANCE OF RANDOM VARIABLE X
1.15: PRACTICAL EXAMPLE FIVE
1.16: PROPERTIES OF VARIANCE
1.17: TOTAL PROBABILITY IN APPLICATION TO EXPECTATION
1.18: CONDITIONAL PROBABILITY
1.19: PRACTICAL EXAMPLE SIX
1.20: PRACTICAL EXAMPLE SEVEN
1.21: PRACTICAL EXAMPLE EIGHT
1.22: PRACTICAL EXAMPLE NINE
1.23: PRACTICAL EXAMPLE TEN
1.24: TEST YOURSELF
CHAPTER TWO
2.0: PROBABILITY DISTRIBUTION
2.1: BINOMIAL DISTRIBUTION
2.2: PRACTICAL EXAMPLE ELEVEN
2.3: EXPECTED VALUE OF BINOMIAL DISTRIBUTION
2.4: VARIANCE OF BINOMIAL DISTRIBUTION
2.5: POISON DISTRIBUTION
2.6: EXPECTED VALUE OF POISON DISTRIBUTION
2.7: VARIANCE OF POISON DISTRIBUTION
2.8: PRACTICAL EXAMPLE TWELVE
2.9: THE GAMMA DISTRIBUTION
2.10: THE EXPECTATION AND CONDITIONS OF GAMMA DISTRIBUTION
2.11: THE EXPECTED VALUE OF GAMMA DISTRIBUTION
2.12: VARIANCE OF GAMMA DISTRIBUTION
2.13: CONDITIONS UNDER WHICH GAMMA CAN TURN TO CHI-SQUARE
2.14: CONDITIONS UNDER WHICH GAMMA CAN TURN TO EXPONENTIAL
2.15: BERNOULI DISTRIBUTION AND ITS EXPECTED VALUE
2.16: VARIANCE OF BERNOULI DISTRIBUTION
2.17: GEOMETRIC DISTRIBUTION
2.18: PRACTICAL EXAMPLE THIRTEEN
2.19: EXPECTED VALUE OF GEOMETRIC DISTRIBUTION
2.20: THE NORMAL DISTRIBUTION
2.21: EXPECTED VALUE OF NORMAL DISTRIBUTION
2.22: VARIANCE OF NORMAL DISTRIBUTION
2.23: THE TRAITS OF NORMAL DISTRIBUTION
2.24: STANDARDIZATION OF NORMAL DISTRIBUTION
2.25: VARIANCE OF STANDARD NORMAL DISTRIBUTION
2.26: EVALUATION AND GEOMETRIC INTERPRETATION OF NORMAL PROBABILITIES.
2.27: PRACTICAL EXAMPLE FOURTEEN
2.28: PRACTICAL EXAMPLE FIFTEEN
2.29: PRACTICAL EXAMPLE SIXTEEN
2.30: BINOMIAL APPROXIMATION TO NORMAL
2.30: PRACTICAL EXAMPLE SEVENTEEN
2.31: PRACTICAL EXAMPLE EIGHTEEN
2.32: USING NORMAL APPROXIMATION TO TREAT BINOMIAL PROBLEMS
CHAPTER THREE
3.1: PROBABILITY: A CONCISE ILLUSTRATION
3.2: LAWS OF PROBABILITY
3.2.0: TEST YOURSELF
3.2.1: LAW 1: BASIC DEFINITIVE LAW OF PROBABILITY
3.2.1.1: PRACTICAL EXAMPLE NINETEEN
3.2.1.2: PRACTICAL EXAMPLE TWENTY
3.2.1.3: PRACTICAL EXAMPLE TWENTY-ONE
3.2.1.3.1: TEST YOURSELF
3.2.1.4: PRACTICAL EXAMPLE TWENTY-TWO
3.2.1.5: PRACTICAL EXAMPLE TWENTY-THREE
3.2.2: LAW 2: ADDITIVE LAW OF PROBABILITY
3.2.2.1: FIRST ADDITION LAW OF PROBABILITY
3.2.2.1.1: PRACTICAL EXAMPLE TWENTY-THREE
3.2.2.2: SECOND ADDITION LAW OF PROBABILITY
3.2.2.2.1: SUMMARIZING THE ADDITIVE RULE OF PROBABILITY
3.2.3: LAW 3: THE MULTIPLICATIVE LAW OF PROBABILITY
3.2.3.1: PRACTICAL EXAMPLE TWENTY-FOUR
3.3: PROBABILITY INVOLVING GAME
3.3.1: PROBABILITY INVOLVING CARD GAME
3.3.1.1: PRACTICAL EXAMPLE TWENTY-FIVE
3.3.2: PROBABILITY INVOLVING LUDO GAME
3.3.2.1: PRACTICAL EXAMPLE TWENTY-SIX
3.3.3: MISCELLANEOUS EXAMPLE
3.3.3.1: MISCELLANEOUS EXAMPLE ONE
3.3.3.2: MISCELLANEOUS EXAMPLE TWO
3.3.3.3: MISCELLANEOUS EXAMPLE THREE
CHAPTER FOUR
4.1: TEST EXERCISES
4.1.1: EASY
4.1.2: MEDIUM
4.1.3: HARD
Objectives and Topics
The primary aim of this work is to provide a comprehensive, accessible guide to the theory and practice of probability. It seeks to demystify complex mathematical concepts for students of varying learning speeds, specifically targeting those who may experience arithmophobia, while also equipping high-level learners and instructors with practical forecasting and predictive skills.
- Foundational terminologies and conceptualization of probability theories
- Statistical probability distributions including Binomial, Poisson, Normal, and Gamma
- Calculation of expected values, variances, and standard deviations
- Application of probability laws for games (cards and Ludo) and real-world scenarios
- Exercises ranging from simple to advanced complexity levels
Excerpt from the Book
1.1: THEORY OF PROBABILITY: THE DOCTRINE OF CHANCES
The word "probability" comes from the Latin word "probabilitatem," which is directly derived from the ancient French word "probabilite," which dates back to the 14th century. Nonetheless, the mathematical definition of probability dates back to 1718, and by the 18th century, it had come to use the word "chance."Thus, the theory of probability was referred to as 'the doctrine of chances', which originated from the Latin word 'cadentia' meaning 'a fall case. It was Abraham De Moivres that first came up with this theory. In general, French mathematicians Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) developed the methodical and mathematical approach to the theory of probability in the middle of the 17th century while attempting to resolve a problem involving the sharing of a stake in an unfinished gambling match that was presented by renowned French gambler "Chevalier de mere" in 1654. This explains why Pierre de Fermat and Blaise Pascal are regarded as the fathers of probability. Thus, this explanation suggests that the theory of probability originated in the gambling game. Probability is therefore a byproduct of gambling. To put it literally, probability refers to the chance or likelihood that a certain event will take place.
Chapter Summary
CHAPTER ONE: Provides an introduction to the origins of probability and defines core terminology, including experiments, random variables, sample spaces, expectation, and variance with practical examples.
CHAPTER TWO: Explores various probability distributions such as Binomial, Poisson, Gamma, Geometric, and Normal, detailing their calculation methods for expected values and variances.
CHAPTER THREE: Details the fundamental laws of probability, including additive and multiplicative rules, and their application to specific board and card games using illustrative examples.
CHAPTER FOUR: Offers a structured collection of test exercises graded by difficulty (easy, medium, hard) to reinforce the understanding of probability theories and distributions covered in previous chapters.
Keywords
Probability, Doctrine of Chances, Random Variable, Sample Space, Expected Value, Variance, Binomial Distribution, Poisson Distribution, Gamma Distribution, Normal Distribution, Standard Deviation, Conditional Probability, Multiplication Rule, Arithmophobia, Probability Density Function
Frequently Asked Questions
What is the primary focus of this textbook?
The book focuses on providing both the theoretical foundations and practical applications of probability theory, designed to be accessible even to readers who struggle with mathematics.
Which key topics does the work cover?
The work covers basic probability terminology, various theoretical distributions, mathematical laws of probability, and their application to real-world scenarios and games.
What is the academic goal of this publication?
The aim is to help students overcome their fear of arithmetic while providing advanced learners and educators with a clear guide to complex prediction and forecasting models.
What scientific methodology is utilized?
The text utilizes mathematical derivations and statistical modeling, simplified through step-by-step practical examples to ensure comprehensibility.
What content is discussed in the main body?
The main body is divided into three descriptive chapters that progress from elementary terminologies to complex distribution functions and, finally, to the laws of probability in game scenarios.
Which keywords characterize this textbook?
Key terms include Probability, Random Variable, Expectation, Variance, Binomial Distribution, Normal Distribution, and the Laws of Probability.
How does the author handle probability in games?
The author uses card games and Ludo as practical case studies to demonstrate binomial events and the application of addition and multiplication rules in a tangible format.
Does the book address the difficulty of learning mathematics?
Yes, the author specifically addresses "arithmophobia" and caters to learners with varying speeds, using simplified explanations and structured exercises.
How are professional applications of probability integrated?
Through practical examples, such as financial project assessment (Project A vs. Project B), the author shows how probability theory informs decision-making in professional fields.
- Quote paper
- Ugwuja Chinonso Oliver (Author), 2024, Probabilitatem Extraordinaire. Theory and Practice of Probability, Munich, GRIN Verlag, https://www.grin.com/document/1515179
