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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2024
Learn how to calculate your chances with easy-to-understand explanations of probability
Probability—the likelihood or chance of an event occurring—is an important branch of mathematics used in business and economics, finance, engineering, physics, and beyond. We see probability at work every day in areas such as weather forecasting, investing, and sports betting. Packed with real-life examples and mathematical problems with thorough explanations, Probability For Dummies helps students, professionals, and the everyday reader learn the basics. Topics include set theory, counting, permutations and combinations, random variables, conditional probability, joint distributions, conditional expectations, and probability modeling. Pass your probability class and play your cards right, with this accessible Dummies guide.
- Understand how probability impacts daily life
- Discover what counting rules are and how to use them
- Practice probability concepts with sample problems and explanations
- Get clear explanations of all the topics in your probability or statistics class
Probability For Dummies is the perfect Dummies guide for college students, amateur and professional gamblers, investors, insurance professionals, and anyone preparing for the actuarial exam.
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Probability For Dummies®
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Table of Contents
Cover
Title Page
Copyright
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: The Certainty of Uncertainty: Probability Basics
Chapter 1: The Probability in Everyday Life
Figuring Out What Probability Means
Coming Up with Probabilities
Probability Misconceptions to Avoid
Chapter 2: Coming to Terms with Probability
A Set Notation Overview
Probabilities of Events Involving A and/or B
Understanding and Applying the Rules of Probability
Recognizing Independence in Multiple Events
Including Mutually Exclusive Events
Distinguishing Independent from Mutually Exclusive Events
Getting Some Practice
Solutions
Chapter 3: Picturing Probability: Venn Diagrams, Tree Diagrams, and Bayes’ Theorem
Diagramming Probabilities with Venn Diagrams
Mapping Out Probabilities with Tree Diagrams
The Law of Total Probability and Bayes’ Theorem
Getting Some Practice
Solutions
Part 2: Counting on Probability and Betting to Win
Chapter 4: Setting the Contingency Table with Probabilities
Organizing a Contingency Table
Finding and Interpreting Probabilities within a Contingency Table
Checking for Independence of Two Events
Getting Some Practice
Solutions
Chapter 5: Applying Counting Rules with Combinations and Permutations
Counting on Permutations
Counting Combinations
Getting Some Practice
Solutions
Chapter 6: Against All Odds: Probability in Gaming
Knowing Your Chances: Probability, Odds, and Expected Value
Playing the Lottery
Hitting the Slot Machines
Spinning the Roulette Wheel
Getting Your Chance to Yell “Bingo!”
Knowing What You’re Up Against: Gambler’s Ruin
The Famous Birthday Problem
Part 3: From A to Binomial: Basic Probability Models
Chapter 7: Probability Distribution Basics
The Probability Distribution of a Discrete Random Variable
Finding and Using the Cumulative Distribution Function
Expected Value, Variance, and Standard Deviation of a Discrete Random Variable
Outlining the Discrete Uniform Distribution
Getting Some Practice
Solutions
Chapter 8: Juggling Success and Failure with the Binomial Distribution
Recognizing the Binomial Model
Finding Probabilities for the Binomial
Formulating the Expected Value and Variance of the Binomial
Dealing with Large n
Getting Some Practice
Solutions
Chapter 9: The Normal (but Never Dull) Distribution
Charting the Basics of the Normal Distribution
Finding and Using Probabilities for a Normal Distribution
Handling Backwards Normal Problems
Getting Some Practice
Solutions
Chapter 10: Approximating a Binomial with a Normal Distribution
Identifying When You Need to Approximate Binomials
Why the Normal Approximation Works When n Is Large Enough
Understanding the Normal Approximation to the Binomial
Approximating a Binomial Probability with the Normal: A Coin Example
Getting Some Practice
Solutions
Chapter 11: Sampling Distributions and the Central Limit Theorem
Surveying a Sampling Distribution
Gaining Access to Your Statistics through the Central Limit Theorem
The Sampling Distribution of the Sample Total
The Sampling Distribution of the Sample Mean,
The Sampling Distribution of the Sample Proportion,
Getting Some Practice
Solutions
Chapter 12: Estimating and Doing Hypothesis Tests with Probability
Confidence Intervals and Probability
Probability and Hypothesis Testing
Getting Some Practice
Solutions
Part 4: Taking It Up a Notch: Advanced Probability Models
Chapter 13: Working with the Poisson (a Nonpoisonous) Distribution
Counting On Arrivals with the Poisson Model
Determining Probabilities for the Poisson
Identifying the Expected Value and Variance of the Poisson
Changing Units over Time or Space: The Poisson Process
Approximating a Poisson with a Normal
Getting Some Practice
Solutions
Chapter 14: Covering All the Angles of the Geometric Distribution
Shaping Up the Geometric Distribution
Finding Probabilities for the Geometric Using the pmf
Uncovering the Expected Value and Variance of the Geometric
Getting Some Practice
Solutions
Chapter 15: Making a Positive out of the Negative Binomial Distribution
Recognizing the Negative Binomial Model
Formulating Probabilities for the Negative Binomial
Exploring the Expected Value and Variance of the Negative Binomial
Getting Some Practice
Solutions
Chapter 16: Remaining Calm about the Hypergeometric Distribution
Zooming In on the Conditions for the Hypergeometric Model
Finding Probabilities for the Hypergeometric Model
Measuring the Expected Value and Variance of the Hypergeometric
Getting Some Practice
Solutions
Part 5: For the Hotshots: Continuous Probability Models
Chapter 17: Staying in Line with the Continuous Uniform Distribution
Understanding the Continuous Uniform Distribution
Determining the Density Function for the Continuous Uniform Distribution
Finding Probabilities for the Continuous Uniform Distribution
Corralling Cumulative Probabilities Using F(x)
Figuring the Expected Value and Variance of the Continuous Uniform
Getting Some Practice
Solutions
Chapter 18: The Exponential (and Its Relationship to Poisson) Exposed
Identifying the Density Function for the Exponential
Determining Probabilities for the Exponential
Figuring Formulas for the Expected Value and Variance of the Exponential
Relating the Poisson and Exponential Distributions
Getting Some Practice
Solutions
Part 6: The Part of Tens
Chapter 19: Ten Steps to a Better Probability Grade
Get into the Problem
Understand the Question
Organize the Information
Write Down the Formula You Need
Check the Conditions
Calculate with Confidence
Show Your Work
Check Your Answer
Interpret Your Results
Make a Review Sheet
Chapter 20: Top Ten (or So) Probability Mistakes
Forgetting That a Probability Must Be between Zero and One
Misinterpreting Small Probabilities
Using Probability for Short-Term Predictions
Thinking That 1-2-3-4-5-6 Can’t Win
“Keep ’Em Coming … I’m on a Roll!”
Giving Every Situation a 50-50 Chance
Switching Conditional Probabilities Around
Applying the Wrong Probability Distribution
Leaving Probability Model Conditions Unchecked
Confusing Permutations and Combinations
Assuming Independence
Appendix: Tables for Reference
Binomial Table
Normal Table
Poisson Table
Index
About the Author
Advertisement Page
Connect with Dummies
End User License Agreement
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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Probability For Dummies®, 2nd Edition
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
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Library of Congress Control Number: 2024947439
ISBN 978-1-394-28188-6 (pbk); ISBN 978-1-394-28190-9 (ebk); ISBN 978-1-394-28189-3 (ebk)
Introduction
Probability is all around you every day — in every decision you make and in everything that happens to you — yet it can’t ever give you a guarantee, which forces you to carry your umbrella and get a flu shot every year “just in case.” A probability question can be so easy to ask, yet so hard to answer. I suppose that’s the beauty as well as the curse of probability. You’re walking through an airport three states away from your home, and you see someone you knew from high school and say, “What are the odds of that happening?” Or you hear about someone who won the lottery not once, but twice, and you wonder if you can have the same luck. Or maybe you just heard your teacher say that the chance of two people in the class having the same birthday is 80 percent, and you think, “No way can that be true — she must be crazy!” Well, before you send your professor to the loony bin, know this: Probability and intuition don’t mix. But don’t worry — this book is here to help.
Foolish Assumptions
I wrote this book for anyone who wants and/or needs to know about probability with little or no experience necessary. If you’re a student, you may be taking a course just in probability, so you’re likely interested in getting help with counting rules, permutations, combinations, and some of the more advanced probability distributions such as the geometric and negative binomial.
Or you may be taking a probability and statistics class, which involves an equal treatment of both probability and statistics. This book helps you with the probability part and Statistics For Dummies, also by yours truly (published by Wiley), helps you with the statistics part. But this book also helps you see how statistics fits into the area of probability, and vice versa. (If you’re taking a straight statistics course, you’re likely to run into more probability than you may have bargained for. If so, this book is for you as well.)
Perhaps you’re interested in probability from an everyday point of view. If so, you can find plenty of real-world information in this book that you’ll find helpful, such as how to find basic probability, win the lottery, become rich and famous, and the like.
Icons Used in This Book
I use various icons in this book to draw your attention to certain features that occur on a regular basis. Think of the icons as road signs you encounter on a trip. Some signs tell you about shortcuts, and others offer more information that you may need; some signs alert you to possible warnings, and others leave you with something to remember.
I use this icon to point out exciting and perhaps surprising situations where people use probability in the real world, from actuarial science to manufacturing (and casinos, of course).
I use this icon for ideas that I hope you’ll remember long after you read this book. It mainly refers to actions you can take to help you determine which technique to use in a given probability problem.
Feel free to skip over the paragraphs that feature this icon if you’re in an introductory-level course or you’re just interested in getting the info you need. Anything marked with this icon is either ancillary or more advanced than is necessary for an introductory probability course. However, if you’re interested in the gory details, or if you have to be for your more advanced-level course, go for it!
This icon flags helpful hints, ideas, or shortcuts that you can use to save time. They may also give you alternative ways to think about a particular concept.
This icon alerts you to specific ways that you may get tripped up working a certain kind of problem. I also use this icon to discuss common misconceptions about probability that can get you into trouble.
Beyond the Book
In addition to the material in the book you’re reading right now, this product also comes with some access-anywhere goodies on the web. Check out this book’s online Cheat Sheet for basic rules of probability and discrete and continuous probability distributions. Just go to www.dummies.com and type Probability For Dummies Cheat Sheet in the Search box.
Where to Go from Here
I wrote this book in a modular way, meaning you can start anywhere and still understand what’s happening. However, I can make some recommendations about where to start:
If you’re taking a probability or statistics class based in algebra, I recommend starting with
Part 1
to build a basic foundation for probability and how to set up problems.
If you’re taking a probability class based in calculus, you may want to start with
Part 4
and work your way to
Part 5
. In
Part 5
, you have a chance to see your calculus at work as you find probabilities as areas under a curve.
If you’re taking a statistics and/or probability course that focuses heavily on counting rules, combinations, and permutations, head to
Chapter 5
. There you’ll find examples of counting problems under every scenario I could think of to help you build a strong set of strategies so each problem doesn’t look different.
If you’re interested in games of chance, head to
Chapters 5
and
6
. You’ll find some ideas on what your expected winnings are with various games, and you’ll discover how to calculate your odds of winning.
Part 1
The Certainty of Uncertainty: Probability Basics
IN THIS PART …
In Part 1, you get started with the basics of probability — the terminology, the basic ideas of finding a probability, and, perhaps most important, how to organize and set up all the information you have in order to successfully calculate a probability. You also discover ways in which people use probability in the real world.
But let’s be honest. When it comes to a class that involves probability, is there truly a real world? Maybe, maybe not. Counting the number of ways to pick 3 green balls and 4 red balls from an urn that contains 20 green balls and 30 red balls doesn’t sound all that relevant — and it isn’t. That’s why you won’t see a single urn problem anywhere in this part. However, if you do run across an urn problem in your life, you’ll know how to answer it, using the techniques from Part 1.
Chapter 2
Coming to Terms with Probability
IN THIS CHAPTER
Nailing down the basic definitions and terms associated with probability
Examining how probability relates different events
Solving probability problems with the rules and formulas of probability
Identifying independent and mutually exclusive events
Probability can be challenging to sort out, but understanding the language is a huge help. The first step toward probability success is having a clear knowledge of the terms, the notation, and the different types of probabilities you come across. If you use and understand the terms, notation, and types when working on easy problems, you have an edge from the start when the problems get more complex. This chapter sets you on the right track.
A Set Notation Overview
Probability has its own set of notations, symbols, and definitions that provide a shorthand way of expressing what you want to do. Notation refers to the symbols that you use as shorthand to talk about probability; for example, P(A) means the probability that A will occur. Definition refers to the statistical meanings of the terms used in probability. Every probability problem starts out by defining the information you have and the quantity you’re trying to get, which all comes down to notation and terms.
Noting outcomes: Sample spaces
A probability is the chance that a certain outcome, or result, will occur out of all the possible outcomes for some random process at hand. The process is called a random process because you conduct an experiment, or other form of data collection, and you don’t know how the results will come out. Before you can figure out the probability of the result you’re interested in, you list all the possible outcomes; this list is called the sample space and is typically denoted by S.
Any collection of items in probability is called a set. Notice that S is a set, so you use set notation to list its outcomes and probabilities for those outcomes (such as using brackets around the list with commas that separate each outcome).
For example, if your random process is rolling a single die, denotes the sample space. The set S can take on three different types: finite, countably infinite, and uncountably infinite.
Finite samples spaces
If you can write and count all the elements in a set, the set is finite.
