An Introduction to Probability: A Concise Exploration of Core Concepts

By Y. Mathew

An Introduction to Probability: A Concise Exploration of Core Concepts highlights the fact that the mathematical notion of Probability relies on ratios to give a numeric value to the level of certainty we can have about a particular outcome for an event. As such, the mathematical concept of ratios or fractions, part-whole relationships, is used to begin the exploration of Probability. The book then goes on to explain in simple, direct language, with minimal reliance on complex technical machinery, how to build sample spaces and develop ratios to predict the probability of a selected outcome for an event. An Introduction to Probability: A Concise Exploration of Core Concepts is a reader-friendly exploration of probability. My approach is unique in that I provide extensive verbal explanations of the basic ideas and concepts which underpin Probability with minimal reliance on the usual technical language of Mathematics consisting of symbols and formulae. The text is written to be a gentle, thoughtful, perhaps even playful, exploration of the basic ideas in Probability. This approach is fueled by my desire to explain - not exclusively to present. I think most math books tend to present the material with very sparse or no detailed verbal explanation. In my book, the emphasis is placed on verbally explaining the basic ideas in Probability. I hope the reader finds this approach helpful. 

• Language: English
• Publisher: Outskirts Press
• Release date: Nov 19, 2023
• ISBN: 9781977270405

Y. Mathew

Y. Mathew presents a stunning new approach to Mathematics with her first published book An Introduction to Probability: A Concise Exploration of Core Concepts. With undergraduate and graduate degrees in several disciplines and a trained teacher with years spent teaching students in a variety of settings, she brings a unique perspective to Mathematics and Mathematics education. 

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An Introduction to Probability: A Concise Exploration of Core Concepts

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Dedication

To God,

whose love, presence, and guidance,

made this book possible.

with love.

Preface

This book is an outgrowth of time spent playing on the IXL website. The colors, images, and enthusiastic congratulatory messages after each successful effort served to re-animate in me a child-like approach to mathematical topics and concepts. Thanks to IXL, I am able to lose myself for a while engaging mathematical topics and concepts in a context where learning and play are equally important. A great big thank you to everyone at IXL - the mathematicians, the educators and curriculum developers, the artists, the programmers and all the other people who work everyday to make IXL a wonderful learning environment. I would like to make clear, however, that I am in no way affiliated with IXL or any of its allied group of companies. This book is not an IXL product and is not intended to imply any sort of integrations with the IXL math curriculum and in particular its coverage of probability topics. This book is my own work and a result of personal reflection on the ideas encountered during my review of Probability at IXL. IXL problem sets are found at the end of each section in the book. They have been included to support the preceding learning material and the problems are arranged in order of difficulty or complexity. These problem sets serve several functions: 1) they provide an opportunity to explore problems based on the ideas discussed in the section; 2) they allow the reader to explore various problem types on this material; 3) lastly, these problem sets serve as footnotes and reference guides to locate and explore the concepts and problems as they appear at IXL. Again, I want to re-state that I have not been paid to endorse IXL nor am I affiliated with IXL or its subsidiaries in any way; however, as a former middle school and high school math teacher and tutor, I believe the site is a valuable learning resource for teachers and students.

As an introductory text on Probability, this book is distinctive because it places a great deal of emphasis on the explanation of mathematical concepts. Not only are core ideas developed clearly in great depth and detail, much time is spent on the careful study of all formulae presented in the text. Hopefully, this book will be an accessible introduction to Probability and provide the reader with a firm grounding in basic probability concepts.

Introduction

Audience

An Introduction to Probability: A Concise Exploration of Core Concepts is written to be helpful to learners in a variety of contexts. As such, the book is not written as a simple linear progression of ideas where an idea is presented once and then never re-visited. I adopt a didactic writing style which is recursive, or spiraling, in nature. As a consequence, I constantly re-visit, review, and reinforce ideas presented earlier. My goal in doing this is to highlight important concepts and to promote comprehension and retention.

This book can be used independently or it can be used to supplement or support existing learning materials and contexts. The book is to be read from start to finish and has been written so that the material builds towards cumulative learning goals. In total, learning is reinforced using four different strategies:

Final Thoughts serve to tie together important ideas and make important observations or points.

Summary tables contain key vocabulary and concepts found in each section.

Questions (with answers provided).

Problem sets drawn from IXL

Students who are in an introductory Probability class can benefit from reading through each chapter and working through the suggested problems. Learners in a work context may find it sufficient to read the book from start to finish. Both sets of learners will appreciate the detailed and approachable explanations of core Probability concepts.

Teachers may find this book helpful as a personal refresher on those areas of Probability required by the Common Core Standards of Learning for the K - 12 curriculum. In addition, the book can serve as a classroom resource either for enrichment for students who have completed classroom instructional materials or as primary reading for the whole class.

Alternatively, teachers can also opt to scan the book for just those topics required for their class and use the book to find problems on IXL relevant to these topic. Finally, teachers can use this book as a coaching guide for students at different learning levels. The problem sets can accommodate learners at different levels and allow each learner to make satisfying progress on their own terms.

Casual readers who are curious about Probability and are looking for a good introduction to subject may find this book valuable for its careful explanation of key concepts. If desired, they can dip into the problem sets to get a sense for how these concepts appear as problems in simplified contexts.

Another word about IXL and the IXL problem sets. While these points have been made earlier, they really must be re-stated because they very important. The IXL topics listed at the end of each section serve two purposes: First, they serve as a supplement to learning. Working out problems really will help to deepen understanding. It is nice to read the examples and explanations, but it is always a good idea to work out a number of problems to really become familiar with a problem type, or a concept, and to allow these to become your own. If you are a student, you may want to have your parent purchase a subscription to IXL in order to have access to sufficient practice problems. However, if you are a casual reader, it is possible to view up to ten sample problems each day as a guest to the site as of this writing. Second, the problem sets serve as references which point the reader to IXL where they will find these concepts and problems as they are presented on IXL.

Overview

The focus in this book has been to explain as concisely as possible, while still being rigorous, the main concepts of introductory Probability. Hopefully, it will enrich readers with a deeper understanding of the subject. The book can be roughly divided into two parts. Chapters 1 through 3 introduce the basic concepts of probability and the eight probability statements. The second part, Chapters 4 through 6, deals with combinations, permutations and the mechanics of constructing combination and permutation sample spaces. This second part of the book can be subdivided into two smaller sections. The first section, Chapters 4 and 5, introduce readers to basic combinatoric concepts. The second section, Chapter 6, shows readers how these concepts are used to create combination and permutation sample spaces as well as derive probability ratios based on these sample spaces. The final chapter and capstone of the book, Chapter 7, provides a summary of the entire book. Below is a brief overview of the book chapter by chapter.

Chapter 1 begins with the question, What is probability? and develops motivations for moving awary from verbal descriptions for the likelihood of an outcome for an event to a rigorous mathematical approach to quantifying the probability of an outcome for an event.

Chapter 2 takes up the question, What is the probability ratio? and explains how a probability ratio, just like other ratios and fractions, compares a part to a whole. In this case, the whole is a set or collection of all known and possible outcomes for an event identified as the sample space for an event. The part represents the number of desired or specified outcomes in the sample space for the event. In addition, the chapter also presents theoretical and experimental probability and looks at how these two are similar and different.

Chapter 3 presents the eight fundamental probability statements. This chapter is divided into two distinct sections. The first section of Chapter 3 looks at single event probability statements. The second section looks at multiple event probability statements. In both sections, the reader will find plenty of approachable examples provided with full explanations.

Chapter 4 introduces readers to basic combinatoric ideas. This chapter begins with a presentation of The Counting Principle and explores how this one principle lies at the heart of computing factorials, permutations, and combinations.

Chapter 5 delves into how combinations and permutations are utilized to create a new sample space from a given sample space. In addition, we learn that when we compute a combination or a permutation in this context, we are actually calculating the size of the new sample space. The new sample space can consist either of combinations or permutations of several outcomes taken, or selected, at a time from the original sample space. The key difference between a combination sample space and a permutation sample space is that a permutation sample space includes all possible selection orders of outcomes while a combination sample space consists exclusively of unique selections of outcomes. For example, the following two selections of outcomes are considered two distinct permutations but considered one combination:

{red, yellow, blue}

{red, blue, yellow}

Insofar as the colors are in different selection orders, both selections will appear in a permutation sample space. However, since both selections contain the same colors, albeit in different selection orders, both selections will not appear in a combination sample space. While this may sound confusing, it will make more sense in the extended discussion on combinations and permutations in Chapter 5.

Chapter 6 picks up where where Chapter 5 ends, and shows readers how the newly created combination or permutation sample space is used to derive the probability of complex outcomes meeting multiple requirements.

Chapter 7 is a review of the entire book. It summarizes key ideas from every chapter in the book and ends with a few final thoughts on probability - its methods, approaches, applications, liabilities, and limitations.

The book ends with two appendices. Appendix A provides additional information on the eight basic probability statements. Appendix B provides additional information on The Counting Principle, Combinations, Permutations, Factorial.

Contents

Preface

Introduction

Chapter 1: What is Probability?

Chapter 2: The Probability Ratio

Chapter 3: The Eight Probability Statements

Chapter 4: Introduction to The Counting Principle, Combinations, Permutations, and Factorial

Chapter 5: Computing Permutations and Combinations

Chapter 6: Using Combinations and Permutations in Probability

Chapter 7: Review

Appendix A: Probability Statements

Appendix B: The Counting Principle, Combinations, Permutations and Factorial

CHAPTER 1

What is Probability?

Key Vocabulary and Concepts

Probability, Likelihood

Event

Verbal vs. Numerical Descriptions of Probability

Probability, as a mathematical discipline, attempts to create a systematic method to quantify the probability or likelihood of an outcome for an event. When we seek the probability of an outcome for an event, we are attempting to verbally describe or numerically quantify the likelihood of a particular outcome for an event.

This raises a prior question, What is an ‘event’? An event is a clearly defined action, state of affairs or condition with a well defined set of possible outcomes.

For example, will it rain tomorrow? The answer to this question can be either a simple ‘yes’ or ‘no’. Alternatively, we can say, ‘There is 10% chance of light showers in the morning…..

The simplest thing we can say about a particular outcome for an event is that it will happen or it will not happen. If we say an outcome will absolutely happen for a given event, we are expressing our certainty that the stated outcome will happen for that event. If desired, we can assign a percentage value to our certainty of a particular outcome for an event and say that we are 100% certain of the identified outcome for an event. In other words, we can say that there is a 100% likelihood of a particular outcome for an event. If we say an outcome will not happen, we are expressing our certainty that a particular outcome for an event will not happen. When we know a particular outcome will not happen or is impossible for an event, we can say that there is a 0% likelihood of said outcome for the event. When we have absolute certainty we can assign a value of 100% or 0% to the probability of an outcome. However, when we are uncertain about the probability of an outcome for an event, what do we do? In the coming pages, we will explore when we can be certain and when uncertainty creeps in.

In order for Probability concepts to be useful, we need to be able to make more fine grained distinctions than 0% or 100%. These are just two ways to describe the likelihood of a particular outcome for an event. In other words, we can say that a particular outcome will happen or a particular outcome will not happen. Below, we will develop a gradation of verbal descriptions for the likelihood of outcomes for events. We start with the simplest statement for a particular outcome of an event: the outcome will happen or it will not happen.

Will Happen -Suppose an outcome for an event is absolutely certain. In other words, we are certain of a particular outcome for an event. In this instance, we say that the probability of this outcome is 100%.

Will Not Happen -Suppose an outcome for an event is impossible. In other words, we are certain that a particular outcome will not occur or is impossible for the event. In this case, we say the probability of this outcome is 0%.

Suppose we have a situation where we are uncertain about the probability of a particular outcome for an event. This outcome may happen or may not happen for the event. In such a situation, we will have the following list of verbal expressions to express our level of certainty.

Will Happen - Suppose an outcome for an event is absolutely certain. In other words, we are certain of a particular outcome for an event. In this instance, we say that the probability of this outcome is 100%.

May or May Not Happen - Suppose we are uncertain about an outcome for an event. It is equally likely that a particular outcome for an event may happen or may not happen. In other words, we are uncertain about an outcome for an event since there is a 50% chance the outcome will occur and a 50% chance the outcome will not occur. Since both circumstances are equally possible we assign a probability value of 50% to the outcome.

Will Not Happen - Suppose an outcome for an event is impossible. In other words, we are certain that a particular outcome will not occur or is impossible for the event. In this case, we say the probability of this outcome is 0%.

Consider a situation where an outcome for an event is quite likely to happen but there is still some degree of uncertainty. In other words, a particular outcome for an event is quite likely to happen but we are not in a position to say it will happen.

Will Happen - Suppose an outcome for an event is absolutely certain. In other words, we are certain of a particular outcome for an event. In this instance, we say that the probability of this outcome is 100%.

Probably Will Happen - Suppose we are fairly certain that an outcome will occur for an event but we cannot rule out that it may not occur. Hence, the likelihood of an outcome for an event happening is more than 50% but less than 100%.

May or May Not Happen -Suppose we are uncertain about an outcome for an event. It is equally likely that a particular outcome for an event may happen or may not happen. In other words, we are uncertain about an outcome for an event since there is a 50% chance the outcome will occur and a 50% chance the outcome will not occur. Since both circumstances are equally possible we assign a probability value of 50% to the outcome.

Will Not Happen - Suppose an outcome for an event is impossible. In other words, we are certain that a particular outcome will not occur or is impossible for the event. In this case, we say the probability of this outcome is 0%.

Similarly, we can say that we are fairly certain an outcome for an event will not happen but we cannot rule out the possibility it may happen. Hence our gradation of verbal descriptors will capture the possibility of an event not happening while recognizing that there is a small chance the outcome may still happen.

Will Happen -Suppose an outcome for an event is absolutely certain. In other words, we are certain of a particular outcome for an event. In this instance, we say that the probability of this outcome is 100%.

Probably Will Happen - It is highly likely or probable that an outcome will occur for an event but we cannot rule out that it may not occur. Hence, the likelihood of an outcome for an event happening is more than 50% but less than 100%.

May or May Not Happen -Suppose we are uncertain about an outcome for an event. It is equally likely that a particular outcome for an event may happen or may not happen. In other words, we are uncertain about an outcome for an