Probability for dummies

Contents at a GlanceIntroduction ...1 Part I: The Certainty of Uncertainty: Probability Basics...7 Chapter 1: The Probability in Everyday Life...9 Chapter 2: Coming to Terms with Probabi

by Deborah Rumsey, PhD

Probability

FOR

FOR

by Deborah Rumsey, PhD

Probability

FOR

Probability For Dummies ®

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About the Author

Deborah Rumsey has a PhD in Statistics from The Ohio State University

(1993) Upon graduating, she joined the faculty in the Department ofStatistics at Kansas State University, where she won the distinguishedPresidential Teaching Award and earned tenure and promotion in 1998

In 2000, she returned to Ohio State and is now a Statistics EducationSpecialist/Auxiliary Faculty Member for the Department of Statistics

Dr Rumsey has served on the American Statistical Association’s StatisticsEducation Executive Committee and is the Editor of the Teaching Bits section

of the Journal of Statistics Education She’s the author of the books Statistics For Dummies and Statistics Workbook For Dummies (Wiley) She also has

published many papers and given many professional presentations on thesubject of Statistics Education Her particular research interests are curricu-lum materials development, teacher training and support, and immersivelearning environments Her passions, besides teaching, include her family,fishing, bird watching, driving a new Kubota tractor on the family “farm,”and Ohio State Buckeye football (not necessarily in that order)

To my husband Eric: Thanks for rolling the dice and taking a chance on me

To my son Clint Eric: Your smile always brings me good luck

Author’s Acknowledgments

Thanks again to Kathy Cox for believing in me and signing me up to write thisbook; to Chrissy Guthrie for her continued excellence and for being a won-derful source of support as my project editor; and to Dr Marjorie Bond,Monmouth College, for another invaluable technical review Thanks to JoshDials for his editing that kept things light Thanks to Kythrie Silva for believ-ing in me; to Peg Steigerwald for her constant support and friendship; and to

my family, especially my parents, for loving me through it all I also wish tothank all the students I have had the privilege of teaching; you are the inspi-ration for all of my work

Publisher’s Acknowledgments

We’re proud of this book; please send us your comments through our Dummies online registration form located at www.dummies.com/register/.

Some of the people who helped bring this book to market include the following:

Acquisitions, Editorial, and Media Development

Project Editor: Christina Guthrie Acquisitions Editor: Kathy Cox Copy Editor: Josh Dials Editorial Program Coordinator: Hanna K Scott Technical Editor: Marjorie Bond, PhD

Editorial Manager: Christine Meloy Beck Editorial Assistants: Erin Calligan, Nadine Bell,

Kristin A Cocks, Product Development Director, Consumer Dummies Michael Spring, Vice President and Publisher, Travel

Kelly Regan, Editorial Director, Travel Publishing for Technology Dummies Andy Cummings, Vice President and Publisher, Dummies Technology/General User Composition Services

Gerry Fahey, Vice President of Production Services Debbie Stailey, Director of Composition Services

Contents at a Glance

Introduction 1

Part I: The Certainty of Uncertainty: Probability Basics 7

Chapter 1: The Probability in Everyday Life 9

Chapter 2: Coming to Terms with Probability 19

Chapter 3: Picturing Probability: Venn Diagrams, Tree Diagrams, and Bayes’ Theorem 39

Part II: Counting on Probability and Betting to Win 65

Chapter 4: Setting the Contingency Table with Probabilities 67

Chapter 5: Applying Counting Rules with Combinations and Permutations 77

Chapter 6: Against All Odds: Probability in Gaming 103

Part III: From A to Binomial: Basic Probability Models 129

Chapter 7: Probability Distribution Basics 131

Chapter 8: Juggling Success and Failure with the Binomial Distribution 151

Chapter 9: The Normal (but Never Dull) Distribution 167

Chapter 10: Approximating a Binomial with a Normal Distribution 187

Chapter 11: Sampling Distributions and the Central Limit Theorem 201

Chapter 12: Investigating and Making Decisions with Probability 221

Part IV: Taking It Up a Notch: Advanced Probability Models 233

Chapter 13: Working with the Poisson (a Nonpoisonous) Distribution 235

Chapter 14: Covering All the Angles of the Geometric Distribution 251

Chapter 15: Making a Positive out of the Negative Binomial Distribution 261

Chapter 16: Remaining Calm about the Hypergeometric Distribution 273

Part V: For the Hotshots: Continuous Probability Models 283

Chapter 17: Staying in Line with the Continuous Uniform Distribution 285

Chapter 18: The Exponential (and Its Relationship to Poisson) Exposed 299

Part VI: The Part of Tens 311

Chapter 19: Ten Steps to a Better Probability Grade 313

Chapter 20: Top Ten (Plus One) Probability Mistakes 323

Appendix: Tables for Your Reference 333

Index 343

Table of Contents

Introduction 1

About This Book 1

Conventions Used in This Book 2

What You’re Not to Read 2

Foolish Assumptions 3

How This Book Is Organized 3

Part I: The Certainty of Uncertainty: Probability Basics 3

Part II: Counting on Probability and Betting to Win 4

Part III: From A to Binomial: Basic Probability Models 4

Part IV: Taking It Up a Notch: Advanced Probability Models 4

Part V: For the Hotshots: Continuous Probability Models 4

Part VI: The Part of Tens 5

Appendix 5

Icons Used in This Book 5

Where to Go from Here 6

Part I: The Certainty of Uncertainty: Probability Basics 7

Chapter 1: The Probability in Everyday Life 9

Figuring Out what Probability Means 9

Understanding the concept of chance 10

Interpreting probabilities: Thinking large and long-term 10

Seeing probability in everyday life 11

Coming Up with Probabilities 12

Be subjective 13

Take a classical approach 13

Find relative frequencies 14

Use simulations 15

Probability Misconceptions to Avoid 17

Thinking in 50-50 terms when you have two outcomes 17

Thinking that patterns can’t occur 18

Chapter 2: Coming to Terms with Probability 19

A Set Notation Overview 19

Noting outcomes: Sample spaces 19

Noting subsets of sample spaces: Events 21

Noting a void in the set: Empty sets 22

Putting sets together: Unions, intersections, and complements 22

Probabilities of Events Involving A and/or B 24

Probability notation 24

Marginal probabilities 25

Union probabilities 26

Intersection (joint) probabilities 26

Complement probabilities 26

Conditional probabilities 27

Understanding and Applying the Rules of Probability 29

The complement rule (for opposites, not for flattering a date) 29

The multiplication rule (for intersections, not for rabbits) 30

The addition rule (for unions of the nonmarital nature) 31

Recognizing Independence in Multiple Events 32

Checking independence for two events with the definition 32

Utilizing the multiplication rule for independent events 33

Including Mutually Exclusive Events 34

Recognizing mutually exclusive events 34

Simplifying the addition rule with mutually exclusive events 35

Distinguishing Independent and Mutually Exclusive Events 36

Comparing and contrasting independence and exclusivity 36

Checking for independence or exclusivity in a 52-card deck 37

Chapter 3: Picturing Probability: Venn Diagrams, Tree Diagrams, and Bayes’ Theorem 39

Diagramming Probabilities with Venn Diagrams 40

Utilizing Venn diagrams to find probabilities beyond those given 40

Using Venn diagrams to organize and visualize relationships 41

Proving intermediate rules about sets, Using Venn diagrams 42

Exploring the limitations of Venn diagrams 44

Finding probabilities for complex problems with Venn diagrams 45

Mapping Out Probabilities with Tree Diagrams 47

Showing multi-stage outcomes with a tree diagram 49

Organizing conditional probabilities with a tree diagram 51

Reviewing the limitations of tree diagrams 54

Drawing a tree diagram to find probabilities for complex events 54

The Law of Total Probability and Bayes’ Theorem 56

Finding a marginal probability using the Law of Total Probability 57

Finding the posterior probability with Bayes’ Theorem 60

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Part II: Counting on Probability and Betting to Win 65

Chapter 4: Setting the Contingency Table with Probabilities 67

Organizing a Contingency Table 67

Defining the sample space 68

Setting up the rows and columns 69

Inserting the data 69

Adding the row, column, and grand totals 70

Finding and Interpreting Probabilities within a Contingency Table 70

Figuring joint probabilities 71

Calculating marginal probabilities 71

Identifying conditional probabilities 72

Checking for Independence of Two Events 74

Chapter 5: Applying Counting Rules with Combinations and Permutations 77

Counting on Permutations 78

Unraveling a permutation 78

Permutation problems with added restrictions: Are we having fun yet? 82

Finding probabilities involving permutations 86

Counting Combinations 88

Solving combination problems 89

Combinations and Pascal’s Triangle 90

Probability problems involving combinations 91

Studying more complex combinations through poker hands 93

Finding probabilities involving combinations 100

Chapter 6: Against All Odds: Probability in Gaming 103

Knowing Your Chances: Probability, Odds, and Expected Value 104

Playing the Lottery 105

Mulling the probability of winning the lottery 105

Figuring the odds 107

Finding the expected value of a lottery ticket 107

Hitting the Slot Machines 111

Understanding average payout 111

Unraveling slot machine myths 113

Implementing a simple strategy for slots 114

Spinning the Roulette Wheel 116

Covering roulette wheel basics 116

Making outside and inside bets 117

Developing a roulette strategy 120

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Getting Your Chance to Yell “BINGO!” 121

Ways to win at BINGO 121

The probability of getting BINGO — more complicated than you may think 123

Knowing What You’re Up Against: Gambler’s Ruin 125

The Famous Birthday Problem 126

Part III: From A to Binomial: Basic Probability Models 129

Chapter 7: Probability Distribution Basics 131

The Probability Distribution of a Discrete Random Variable 131

Defining a random variable 132

Finding and using the probability distribution 133

Finding and Using the Cumulative Distribution Function (cdf) 138

Interpreting the cdf 139

Graphing the cdf 140

Finding probabilities with the cdf 141

Determining the pmf given the cdf 143

Expected Value, Variance, and Standard Deviation of a Discrete Random Variable 144

Finding the expected value of X 145

Calculating the variance of X 147

Finding the standard deviation of X 148

Outlining the Discrete Uniform Distribution 148

The pmf of the discrete uniform 149

The cdf of the discrete uniform 149

The expected value of the discrete uniform 150

The variance and standard deviation of the discrete uniform 150

Chapter 8: Juggling Success and Failure with the Binomial Distribution 151

Recognizing the Binomial Model 151

Checking the binomial conditions step by step 152

Spotting a variable that isn’t binomial 153

Finding Probabilities for the Binomial 155

Finding binomial probabilities with the pmf 155

Finding binomial probabilities with the cdf 160

Formulating the Expected Value and Variance of the Binomial 165

The expected value of the binomial 165

The variance and standard deviation of the binomial 166

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Chapter 9: The Normal (but Never Dull) Distribution 167

Charting the Basics of the Normal Distribution 167

The shape, center, and spread 168

The standard normal (Z) distribution 170

Finding and Using Probabilities for a Normal Distribution 172

Getting the picture 173

Translating a problem into probability notation 173

Using the Z-formula 174

Utilizing the Z table to find the probability 176

Handling Backwards Normal Problems 180

Setting up a backwards normal problem 181

Using the Z table backward 183

Returning to X units, using the Z-formula solved for X .185

Chapter 10: Approximating a Binomial with a Normal Distribution 187

Identifying When You Need to Approximate Binomials 187

Why the Normal Approximation Works when n Is Large Enough 188

Symmetric situations: When p is close to 0.50 189

Skewed situations: When p is close to zero or one 190

Understanding the Normal Approximation to the Binomial 192

Determining if n is large enough 192

Finding the mean and standard deviation to put in the Z-formula 193

Making the continuity correction 194

Approximating a Binomial Probability with the Normal: A Coin Example 197

Chapter 11: Sampling Distributions and the Central Limit Theorem 201

Surveying a Sampling Distribution 202

Setting up your sample statistic 202

Lining up possibilities with the sampling distribution 202

Saved by the Central Limit Theorem 204

Gaining Access to Your Statistics through the Central Limit Theorem (CLT) 205

The main result of the CLT 205

Why the CLT works 206

The Sampling Distribution of the Sample Total (t) 210

The CLT applied to the sample total 211

Finding probabilities for t with the CLT 211

The Sampling Distribution of the Sample Mean, X 214

The CLT applied to the sample mean 215

Finding probabilities for X with the CLT 216

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The Sampling Distribution of the Sample Proportion, pt 217

The CLT applied to the sample proportion 217

Finding probabilities for pt with the CLT 218

Chapter 12: Investigating and Making Decisions with Probability 221

Confidence Intervals and Probability 221

Guesstimating a probability 222

Assessing the cost of probably (hopefully?) being right 224

Interpreting a confidence interval with probability 225

Probability and Hypothesis Testing 226

Testing a probability 226

Putting the p in probability with p-values 228

Accepting the probability of making the wrong decision 229

Putting the lid on data snoopers 230

Probability in Quality Control 231

Part IV: Taking It Up a Notch: Advanced Probability Models 233

Chapter 13: Working with the Poisson (a Nonpoisonous) Distribution 235

Counting On Arrivals with the Poisson Model 236

Meeting conditions for the Poisson model 236

Pitting Poisson versus binomial 237

Determining Probabilities for the Poisson 237

The pmf of the Poisson 238

The cdf of the Poisson 240

Identifying the Expected Value and Variance of the Poisson 243

Changing Units Over Time or Space: The Poisson Process 244

Approximating a Poisson with a Normal 245

Satisfying conditions for using the normal approximation 246

Completing steps to approximate the Poisson with a normal 248

Chapter 14: Covering All the Angles of the Geometric Distribution 251

Shaping Up the Geometric Distribution 252

Meeting the conditions for a geometric distribution 252

Choosing the geometric distribution over the binomial and Poisson 253

Finding Probabilities for the Geometric by Using the pmf 254

Building the pmf for the geometric 255

Applying geometric probabilities 256

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Uncovering the Expected Value and Variance of the Geometric 258

The expected value of the geometric 258

The variance and standard deviation of the geometric 259

Chapter 15: Making a Positive out of the Negative Binomial Distribution 261

Recognizing the Negative Binomial Model 261

Checking off the conditions for a negative binomial model 262

Comparing and contrasting the negative binomial, geometric, and binomial models 262

Formulating Probabilities for the Negative Binomial 264

Developing the negative binomial probability formula 264

Applying the negative binomial pmf 265

Exploring the Expected Value and Variance of the Negative Binomial 269

The expected value of the negative binomial 269

The variance and standard deviation of the negative binomial 270

Applying the expected value and variance formulas 271

Chapter 16: Remaining Calm about the Hypergeometric Distribution 273

Zooming In on the Conditions for the Hypergeometric Model 274

Finding Probabilities for the Hypergeometric Model 275

Setting up the hypergeometric pmf 275

Breaking down the boundary conditions for X 277

Finding and using the pmf to calculate probabilities 279

Measuring the Expected Value and Variance of the Hypergeometric 281

The expected value of the hypergeometric 281

The variance and standard deviation of the hypergeometric 281

Part V: For the Hotshots: Continuous Probability Models 283

Chapter 17: Staying in Line with the Continuous Uniform Distribution 285

Understanding the Continuous Uniform Distribution 286

Determining the Density Function for the Continuous Uniform Distribution 287

Building the general form of f(x) 287

Finding f(x) given a and b 288

Finding the value of b given f(x) 289

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Drawing Up Probabilities for the Continuous Uniform Distribution 290

Finding less-than probabilities 291

Finding greater-than probabilities 292

Finding probabilities between two values 293

Corralling Cumulative Probabilities, Using F(x) 294

Figuring the Expected Value and Variance of the Continuous Uniform 296

The expected value of the continuous uniform 297

The variance and standard deviation of the continuous uniform 297

Chapter 18: The Exponential (and Its Relationship to Poisson) Exposed 299

Identifying the Density Function for the Exponential 300

Determining Probabilities for the Exponential 302

Finding a less-than probability for an exponential 302

Finding a greater-than probability for an exponential 304

Finding a between-values probability for an exponential .305

Figuring Formulas for the Expected Value and Variance of the Exponential 307

The expected value of the exponential 307

The variance and standard deviation of the exponential 308

Relating the Poisson and Exponential Distributions 309

Part VI: The Part of Tens 311

Chapter 19: Ten Steps to a Better Probability Grade 313

Get Into the Problem 314

Understand the Question 314

Organize the Information 315

Write Down the Formulas You Need 316

Check the Conditions 317

Calculate with Confidence 318

Show Your Work 319

Check Your Answer 319

Interpret Your Results 321

Make a Review Sheet 321

Chapter 20: Top Ten (Plus One) Probability Mistakes 323

Forgetting a Probability Must Be Between Zero and One 323

Misinterpreting Small Probabilities 324

Using Probability for Short-Term Predictions 325

Thinking That 1-2-3-4-5-6 Can’t Win 325

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“Keep ’em Coming I’m on a Roll!” 326

Giving Every Situation a 50-50 Chance 326

Switching Conditional Probabilities Around 327

Applying the Wrong Probability Distribution 328

Leaving Probability Model Conditions Unchecked 329

Confusing Permutations and Combinations 330

Assuming Independence 331

Appendix: Tables for Your Reference 333

Binomial Table 333

Normal Table .337

Poisson Table 340

Index 343

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Probability For Dummies

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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 tee, which forces you to carry your umbrella and get a flu shot every year “just

guaran-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 walkingthrough an airport three states away from your home, and you see someoneyou 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 youwonder if you could have the same luck Or maybe you just heard your teachersay 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 — he must be crazy!”Well, before you send your professor to the loony bin, know this: Probabilityand intuition don’t mix But don’t worry — this book is here to help

About This Book

The main goal of this book is to cut down the amount of time you spend ning your wheels to figure out a probability The design of this book allowsyou to quickly find out how to solve the probability questions you’re asking(or that you have to answer)

spin-This book gives you the tools to read, set up, and solve a wide range of ability problems Because all probability problems tend to look different, I buildstrategies that help you identify what type of problem you’re working with,what tools you need to pull out to solve it, and what calculations get you thecorrect answer You also gain practice interpreting probability and discoveringwhat misconceptions and common errors you should avoid

prob-Along the way, you find some interesting surprises and a bird’s eye view ofhow probability pulls on the strings of the real world I also include tips andstrategies for playing games of chance, so if you do win the lottery, you canwrite about this book in your travel journal on the way to Fiji!

This book is different from other probability books in many ways:

I focus on material that instructors cover in probability and/or statisticscourses, in addition to real-world probability topics Most probabilitybooks out there help you win casino games but don’t help you out muchwith the probability problems you see in a probability and/or statisticscourse

I provide an extensive number of examples to cover the many differenttypes of problems you face.

You see plenty of tips, strategies, and warnings based on my vast experience with students of all backgrounds and learning styles (and my experiences with grading their papers)

I focus on building strong problem-solving skills to help you develop asimilar problem-solving strategy when you take exams

My nonlinear approach allows you to skip around in the book and stillhave easy access and understanding of any given topic

The conversational narrative comes from a student’s point of view

I use understandable language to help you comprehend, remember, andput into practice probability definitions, techniques, and processes

I concentrate on clear and concise step-by-step procedures that itively explain how to work through probability problems and rememberhow to do them later on

intu-Conventions Used in This Book

In this book, I use the following conventions:

When I introduce and define a new probability-related term,

I italicize it.

The following symbol indicates multiplication: *

What You’re Not to Read

It pains me to tell you that any part of my book is skippable, but I have to

be honest: You can pass right over any paragraphs that I mark with theTechnical Stuff icon, if you’re so inclined, and be no worse for wear

Also, throughout the book you’ll find sidebars (the gray boxes) that containfun and interesting, yet skippable, tidbits I often use these sidebars to illus-trate how people put probability to use in everyday life Taking a moment toread the sidebars will enhance your understanding and appreciation of prob-ability, but if you’re pressed for time or simply uninterested, you won’t missout on any essential information

Foolish Assumptions

I wrote this book for anyone who wants and/or needs to know about ity with little or no experience necessary For students, you may be taking acourse just in probability, and you’re interested in getting help with countingrules, permutations, combinations, and some of the more advanced probabilitydistributions such as the geometric and negative binomial

probabil-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 [Wiley],

helps you with the statistics) But it also helps you see how statistics fits intothe area of probability, and vice versa (If you’re taking a straight statisticscourse, you’re likely to run into more probability than you may have bar-gained 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 findhelpful, such as how to find basic probability, win the lottery, become richand famous, and the like

How This Book Is Organized

This book is organized into five major parts that explore the main topic areas

in probability I also include a part that offers a couple quick top-ten ences for you to use Each part contains chapters that break down each majorobjective into understandable pieces

refer-Part I: The Certainty of Uncertainty:

Probability Basics

This part gives you the fundamentals of probability, along with strageties forsetting up and solving the most common probability problems in the introduc-tory course It starts by introducing probability as a topic that has an impact

on all of us every day and underscores the point that probability often goesagainst our intuition You discover the basic definitions, terms, notation, andrules for probability, and you get answers to those all-important (and oftenfrustrating) questions that perplex students of probability, such as, “What’sthe real difference between independent and mutually exclusive events?”

You also see different methods for organizing the information given to you,including Venn diagrams, tree diagrams, and tables Finally, you discovergood strategies for solving more complex probability problems involving theLaw of Total Probability and Bayes’ Theorem

3

Introduction

Part II: Counting on Probability and Betting to Win

In this part, you get down to the nitty gritty of probability, solving problems thatinvolve two-way tables, permutations and combinations, and games of chance.The bottom line in this part? Probability and intuition don’t always mix!

Part III: From A to Binomial:

Basic Probability Models

In this part, you build an important foundation for creating, using, and ating probability models You discover all the ins and outs of a probabilitydistribution; the basic concepts and rules for defining probability distribu-tions; and how to find probabilities, means, and variances You work with thebinomial and normal distributions, and you find out how probability ties in tothe major results from statistics: the Central Limit Theorem, hypothesis test-ing, and overall decision making in the real world

evalu-Part IV: Taking It Up a Notch:

Advanced Probability Models

In this part, you work with more intermediate probability models that countand try to predict the number of arrivals, successes, or the number of trialsneeded to achieve a certain goal The probability distributions I focus on arethe Poisson, negative binomial, geometric, and hypergeometric You find outhow many customers you expect to come into a bank (Poisson distribution);the number of poker hands you need to draw before you get four of a kind(geometric distribution); the number of frames you need to bowl before get-ting your third strike (the negative binomial distribution); and the probability

of getting a hand in poker (hypergeometric distribution)

Part V: For the Hotshots: Continuous Probability Models

In this part, you look at some of the models you find in probability and tistics courses that have calculus as a prerequisite — mainly the uniform(continuous case) distribution, exponential distribution, and other user-defined probability density functions You see how to find probabilities andthe expected value, variance, and standard deviation of continuous probabil-ity models And you apply the models to situations such as the time between

arrivals of customers at the bank, time to complete a task, or the length of a

phone call Note: Calculus is useful but not required for this part I introduce

the methods that use calculus, but I also provide formulas and other ods of solution that don’t use calculus for the uniform and exponential

meth-Part VI: The meth-Part of Tens

In this part, you find my top tens lists: ten steps to a better probability gradeand ten probability misconceptions and how to avoid them This information

is based on my years of experience teaching, answering questions, writingquestions, and grading homework This part will help you pinpoint the mostimportant ideas in probability and the most common errors that are made Italso serves as a quick and condensed resource as you are studying for exams

Appendix

I also include an appendix that contains three handy tables for your ence These tables help you find probabilities for the binomial distribution,the normal distribution, and the Poisson distribution

refer-Icons Used in This Book

I use various icons in this book to draw your attention to certain features thatoccur on a regular basis Think of the icons as road signs you encounter on atrip Some signs tell you about shortcuts, and others offer more informationthat you may need; some signs alert you to possible warnings, and othersleave you with something to remember

I use this icon to point out exciting and perhaps surprising situations wherepeople use probability in the real world, from actuarial science to manufac-turing (and casinos, of course)

These I save for particular ideas that I hope you’ll remember long after youread this book They mainly refer to actions you can take to help you deter-mine which technique to use in a given probability problem

Feel free to skip over the paragraphs that feature this icon if you’re in anintroductory level course The info is either ancillary or more advanced than

is necessary for an introductory probability course However, if you’re ested in the gory details, or if you have to be for your more advanced levelcourse, go for it!

inter-5

Introduction

Tips refer to 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.

Warning icons alert you to specific ways that you may get tripped up working

a certain kind of problem I also reserve this icon to discuss common ceptions about probability that can get you into trouble

miscon-Where to Go from Here

I wrote this book in a modular way, meaning you can start anywhere and stillunderstand what’s happening However, I can make some recommendations

to people who are unsure about where to start:

If you’re taking a probability or statistics class based in algebra, I mend starting with Part I to build a basic foundation for probability andhow to set up problems

recom-
If you’re taking a probability class based in calculus, you may want tostart with Part IV and work your way to Part V In Part V, you have achance to see your calculus at work as you find probabilities as areasunder 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 Icould think of to help you build a strong set of strategies so each problemdoesn’t look different

If you’re interested in games of chance, head to Chapters 5 and 6 You’llfind some ideas on what your expected winnings are with various games,and you’ll discover how to calculate your odds of winning

Part I

The Certainty

of Uncertainty: Probability Basics

In this part

In Part I, you get started with the basics of probability —the terminology, the basic ideas of finding a probability,and, perhaps most importantly, how to organize and set upall the information you have in order to successfully calcu-late a probability You also discover ways in which peopleuse 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 three green balls andfour red balls from an urn that contains twenty green ballsand thirty 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 I

Chapter 1

The Probability in Everyday Life

In This Chapter

Recognizing the prevalence and impact of probability in your everyday life

Taking different approaches to finding probabilities

Steering clear of common probability misconceptions

You’ve heard it, thought it, and said it before: “What are the odds of thathappening?” Someone wins the lottery not once, but twice You acciden-tally run into a friend you haven’t seen since high school during a vacation inFlorida A cop pulls you over the one time you forget to put your seatbelt on

And you wonder what are the odds of this happening? That’s what this

book is about: figuring, interpreting, and understanding how to quantify therandom phenomena of life But it also helps you realize the limitations ofprobability and why probabilities can take you only so far

In this chapter, you observe the impact of probability on your everyday lifeand some of the ways people come up with probabilities You also find outthat with probability, situations aren’t always what they seem

Figuring Out what Probability Means

Probabilities come in many different disguises Some of the terms people use

for probability are chance, likelihood, odds, percentage, and proportion But the basic definition of probability is the long-term chance that a certain outcome

will occur from some random process A probability is a number betweenzero and one — a proportion, in other words You can write it as a percent-age, because people like to talk about probability as a percentage chance, oryou can put it in the form of odds The term “odds,” however, isn’t exactly the

same as probability Odds refers to the ratio of the denominator of a

probabil-ity to the numerator of a probabilprobabil-ity For example, if the probabilprobabil-ity of a horsewinning a race is 50 percent (1⁄2), the odds of this horse winning are 2 to 1

Understanding the concept of chance

The term chance can take on many meanings It can apply to an individual

(“What are my chances of winning the lottery?”), or it can apply to a group(“The overall percentage of adults who get cancer is ”) You can signify achance with a percent (80 percent), a proportion (0.80), or a word (such as

“likely”) The bottom line of all probability terms is that they revolve aroundthe idea of a long-term chance When you’re looking at a random process(and most occurrences in the world are the results of random processes forwhich the outcomes are never certain), you know that certain outcomes canhappen, and you often weigh those outcomes in your mind It all comes down

to long-term chance; what’s the chance that this or that outcome is going tooccur in the long term (or over many individuals)?

If the chance of rain tomorrow is 30 percent, does that mean it won’t rainbecause the chance is less than 50 percent? No If the chance of rain is

30 percent, a meteorologist has looked at many days with similar conditions

as tomorrow, and it rained on 30 percent of those days (and didn’t rain theother 70 percent) So, a 30-percent chance for rain means only that it’s unlikely

to rain

Interpreting probabilities: Thinking large and long-term

You can interpret a probability as it applies to an individual or as it applies

to a group Because probabilities stand for long-term percentages (see theprevious section), it may be easier to see how they apply to a group ratherthan to an individual But sometimes one way makes more sense than theother, depending on the situation you face The following sections outlineways to interpret probabilities as they apply to groups or individuals so youdon’t run into misinterpretation problems

Playing the instant lottery

Probabilities are based on long-term percentages (over thousands of trials), sowhen you apply them to a group, the group has to be large enough (the largerthe better, but at least 1,500 or so items or individuals) for the probabilities toreally apply Here’s an example where long-term interpretation makes sense inplace of short-term interpretation Suppose the chance of winning a prize in aninstant lottery game is 1⁄10, or 10 percent This probability means that in thelong term (over thousands of tickets), 10 percent of all instant lottery ticketspurchased for this game will win a prize, and 90 percent won’t It doesn’t meanthat if you buy 10 tickets, one of them will automatically win

10 Part I: The Certainty of Uncertainty: Probability Basics

If you buy many sets of 10 tickets, on average, 10 percent of your tickets willwin, but sometimes a group of 10 has multiple winners, and sometimes it has

no winners The winners are mixed up amongst the total population of tickets

If you buy exactly 10 tickets, each with a 10 percent chance of winning, youmight expect a high chance of winning at least one prize But the chance ofyou winning at least one prize with those 10 tickets is actually only 65 percent,and the chance of winning nothing is 35 percent (I calculate this probabilitywith the binomial model; see Chapter 8)

Pondering political affiliation

You can use the following example as an illustration of the limitation of probability — namely that actual probability often applies to the percentage of

a large group Suppose you know that 60 percent of the people in your nity are Democrats, 30 percent are Republicans, and the remaining 10 percentare Independents or have another political affiliation If you randomly selectone person from your community, what’s the chance the person is a Democrat?

commu-The chance is 60 percent You can’t say that the person is surely a Democratbecause the chance is over 50 percent; the percentages just tell you that theperson is more likely to be a Democrat Of course, after you ask the person,

he or she is either a Democrat or not; you can’t be 60-percent Democrat

Seeing probability in everyday life

Probabilities affect the biggest and smallest decisions of people’s lives

Pregnant women look at the probabilities of their babies having certaingenetic disorders Before you sign the papers to have surgery, doctors andnurses tell you about the chances that you’ll have complications And beforeyou buy a vehicle, you can find out probabilities for almost every topic regard-ing that vehicle, including the chance of repairs becoming necessary, of thevehicle lasting a certain number of miles, or of you surviving a front-end crash

or rollover (the latter depends on whether you wear a seatbelt — another factbased on probability)

While scanning the Internet, I quickly found several examples of probabilitiesthat affect people’s everyday lives — two of which I list here:

 Distributing prescription medications in specially designed blister packages rather than in bottles may increase the likelihood that consumers will take the medication properly, a new study suggests.

(Source: Ohio State University Research News, June 20, 2005)

In other words, the probability of consumers taking their medicationsproperly is higher if companies put the medications in the new packagingthan it is when the companies put the medicines in bottles You don’t knowwhat the probability of taking those medications correctly was originally

or how much the probability increases with this new packaging, but you

do know that according to this study, the packaging is having some effect

11

Chapter 1: The Probability in Everyday Life

 According to State Farm Insurance, the top three cities for auto theft

in Ohio are Toledo (580.23 thefts per 100,000 vehicles), Columbus (558.19 per 100,000), and Dayton-Springfield (525.06 per 100,000).

The information in this example is given in terms of rate; the studyrecorded the number of cars stolen each year in various metropolitanareas of Ohio Note that the study reports the information as the number

of thefts per 100,000 vehicles The researchers needed a fixed number ofvehicles in order to be fair about the comparison If the study used onlythe number of thefts, cities with more cars would always rank higherthan cities with fewer cars

How did the researchers get the specific numbers for this study? Theytook the actual number of thefts and divided it by the total number ofvehicles to get a very small decimal value They multiplied that value

by 100,000 to get a number that’s fair for comparison To write therates as probabilities, they simply divided them by 100,000 to putthem back in decimal form For Toledo, the probability of car theft is580.23 ÷100,000 = 0.0058023, or 0.58 percent; for Columbus, the proba-bility of car theft is 0.0055819, or 0.56 percent; and for Dayton-Springfield,the probability is 0.0052506, or 0.53 percent

Be sure to understand exactly what format people use to discuss or report

a probability, and be sure that the format allows for a fair and equitable comparison

Coming Up with Probabilities

You can figure or compute probabilities in a variety of ways, depending

on the complexity of the situation and what exactly is possible to quantify.Some probabilities are very difficult to figure, such as the probability of atropical storm developing into a hurricane that will ultimately make landfall

at a certain place and time — a probability that depends on many elementsthat are themselves nearly impossible to determine If people calculate actualprobabilities for hurricane outcomes, they make estimates at best

Some probabilities, on the other hand, are very easy to calculate for an exactnumber, such as the probability of a fair die landing on a 6 (1 out of 6, or 0.167).And many probabilities are somewhere in between the previous two examples

in terms of how difficult it is to pinpoint them numerically, such as the bility of rain falling tomorrow in Seattle For middle-of-the-road probabilities,past data can give you a fairly good idea of what’s likely to happen

proba-12 Part I: The Certainty of Uncertainty: Probability Basics

After you analyze the complexity of the situation, you can use one of four majorapproaches to figure probabilities, each of which I discuss in this section.

Be subjective

The subjective approach to probability is the most vague and the least entific It’s based mostly on opinions, feelings, or hopes, meaning that youtypically don’t use this type of probability approach in real scientific endeavors

sci-You basically say, “Here’s what I think the probability is.” For example, althoughthe actual, true probability that the Ohio State football team will win thenational championship is out there somewhere, no one knows what it is, eventhough every fan and analyst will have ideas about what that chance is, based

on everything from dreams they had last night, to how much they love or hateOhio State, to all the statistics from Ohio State football over the last 100 years

Other people will take a slightly more scientific approach — evaluating players’

stats, looking at the strength of the competition, and so on But in the end,the probability of an event like this is mostly subjective, and although thisapproach isn’t scientific, it sure makes for some great sports talk amongstthe fans!

Take a classical approach

The classical approach to probability is a mathematical, formula-basedapproach You can use math and counting rules to calculate exact proba-bilities in many cases (for more on the counting rules, see Chapter 5)

Anytime you have a situation where you can enumerate the possible comes and figure their individual probabilities by using math, you can usethe classical approach to getting the probability of an outcome or series ofoutcomes from a random process

out-For example, when you roll two die, you have six possible outcomes for the firstdie, and for each of those outcomes, you have another six possible outcomesfor the second die All together, you have 6 * 6 = 36 possible outcomes for thepair In order to get a sum of two on a roll, you have to roll two 1s, meaning itcan happen in only one way So, the probability of getting a sum of two is 1⁄36.The probability of getting a sum of three is 2⁄36, because only two of the outcomesresult in a sum of three: 1-2 or 2-1 A sum of seven has a probability of 6⁄36, or

1⁄6— the highest probability of any sum of two die Why is seven the sum withhighest probability? Because it has the most possible ways of coming up: 1-6,2-5, 3-4, 4-3, 5-2, and 6-1 That’s why the number seven is so important in thegambling game craps (For more on this example, see Chapter 2.)

13

Chapter 1: The Probability in Everyday Life

You also use the classical approach when you make certain assumptions about

a random process that’s occurring For example, if you can assume that theprobability of achieving success when you’re trying to make a sale is thesame on each trial, you can use the binomial probability model for figuringout the probability of making 5 sales in 20 tries Many types of probabilitymodels are available, and I discuss many of them in this book (For more onthe binomial probability model, see Chapter 8.)

The classical approach doesn’t work when you can’t describe the possibleindividual outcomes and come up with some mathematical way of determin-ing the probabilities For example, if you have to decide between differentbrands of refrigerators to buy, and your criterion is having the least chance

of needing repairs in the next five years, the classical approach can’t helpyou for a couple reasons First, you can’t assume that the probability of arefrigerator needing one repair is the same as the probability of needing two,three, or four repairs in five years Second, you have no math formula tofigure out the chances of repairs for different brands of refrigerators; itdepends on past data that’s been collected regarding repairs

Find relative frequencies

In cases where you can’t come up with a mathematical formula or model tofigure a probability, the relative frequency approach is your best bet Theapproach is based on collecting data and, based on that data, finding the percentage of time that an event occurred The percentage you find is the

relative frequency of that event — the number of times the event occurred

divided by the total number of observations made (You can find the bilities for the refrigerator repairs example in the previous section with therelative frequency approach by collecting data on refrigerator repairrecords.)

proba-Suppose, for example, that you’re watching your birdfeeder, and you notice alot of cardinals coming for dinner You want to find the probability that the nextbird that comes to the feeder is a cardinal You can estimate this probability

by counting the number of birds that come to your feeder over a period oftime and noting how many cardinals you see If you count 100 bird visits, and

27 of the visitors are cardinals, you can say that for the period of time youobserve, 27 out of 100 visits — or 27 percent, the relative frequency — weremade by cardinals Now, if you have to guess the probability that the next bird

to visit is a cardinal, 27 percent would be your best guess You come up with

a probability based on relative frequency

14 Part I: The Certainty of Uncertainty: Probability Basics

A limitation of the relative frequency approach is that the probabilities youcome up with are only estimates because you base them on finite samples ofdata you collect And those estimates are only as good as the data that youcollect For example, if you collected your birdfeeder data when you offeredsunflower seeds, but now you offer thistle seed (loved by smaller birds), yourprobability of seeing a cardinal changes Also, if you look at the feeder only at

5 p.m each day, when cardinals are more likely to be out than any other bird,your predictions work only at that same time period, not at noon when all thefinches are also out and about The issue of collecting good data is a statisti-

cal one; see Statistics For Dummies (Wiley) for more information.

 You create the data (usually with a computer); you don’t collect it out inthe real world

 The amount of data is typically much larger than the amount you couldobserve in real life

 You use a certain model that scientists come up with, and models haveassumptions

15

Chapter 1: The Probability in Everyday Life

Consuming data with Consumer Reports

The magazine Consumer Reports — put out by the

Consumers Union, a nonprofit group that helpsprovide consumer protection information — doesthousands of studies to test different makes andmodels of products so it can report on how safe,reliable, effective, and efficient the models are,along with how much they cost In the end, thegroup comes up with a list of recommendationsregarding which models are the best values for

your money Consumer Reports bases its reports

on a relative frequency approach For example,when comparing refrigerators, it tests variousmodels for energy efficiency, temperature per-formance, noise, ease of use, and energy costper year The researchers figure out what per-centage of time the refrigerators need repairs,don’t perform properly, and so on, and they basetheir reports on what they find

You can see an example of a simulation if you let a computer play out a game

of chance for you You can tell it to credit you with $1.00 if a head comes

up on a coin flip and deduct $1.00 if a tail comes up Repeat the bet sands of times and see what you end up with Change the probabilities ofheads and tails to see what happens Your experiments are examples ofsimple simulations

thou-One commonality between simulations and the relative frequency approach

is that your results are only as good as the data you come up with I ber very clearly a simulation that a student performed to predict which teamwould win the NCAA basketball tournament some years ago The student gaveeach of the 64 teams in the tournament a probability of winning its game based

remem-on certain statistics that the sports gurus came up with The student fedthose probabilities into the computer and made the computer repeat thetournament over and over millions of times, recording who won each gameand who won the entire tournament On 96 percent of the simulations, DukeUniversity won the whole thing So, of course, it seemed as if Duke was ashoe-in that year Guess how long Duke actually lasted? The team went down

in the second of six rounds

16 Part I: The Certainty of Uncertainty: Probability Basics

Tracking down hurricanes

One major area where professionals use puter models is in predicting the arrival, intensity,and path of tropical storms, including hurricanes

com-Computer hurricane models help scientists andleaders perform integrated cost-benefit studies;

evaluate the effects of regulatory policies; andmake decisions during crises Insurance compa-nies use the models to make predictions regard-ing the number of and estimated damage due tofuture hurricanes, which helps them adjust theirpremiums appropriately to be ready to pay outthe huge claims that come with large hurricanes

Computer models for tropical storms are best atpredicting long-run (versus short-term) lossesacross large (versus small) geographic areas,

due to the high margin of error Margin of error

is the amount by which your results areexpected to change from sample to sample Youcan’t look at a single storm and say exactly what’sgoing to happen AIR Worldwide, whose com-puter models are used by half the residential

property insurance markets in Florida and

85 percent of the companies that underwriteinsurers, calculates projections over stormsacross a 50,000-year span Another modelingexpert recently lengthened its computer model-ing from 100,000 to 300,000 years to get resultswithin an acceptable margin of error

The models contain so many variables that it takesmany trials to approach a predictable average.Flipping a coin, for instance, has only one variablewith two outcomes If you want to estimate theprobability of flipping heads by using a model, ittakes about 2,500 trials to get a result within a2-percent margin of error The more variables, themore trials required to get a dependable outcome.And with hurricanes, the number of variables ishuge The computer models used by the NationalHurricane Center include variables such as theinitial latitude and longitude of the storm, thecomponents of the “storm motion vector,” andthe initial storm intensity, just to name a few

Probability Misconceptions to Avoid

No matter how researchers calculate a probability or what kind of tion or data they base it on, the probability is often misinterpreted or applied

informa-in the wrong way by the media, the public, and even other researchers whodon’t quite understand the limitations of probability The main idea is thatprobability often goes against your intuition, and you have to be very carefulabout not letting your intuition get the better of you when thinking in terms

of probability This section highlights some of the most common tions about probability

misconcep-Thinking in 50-50 terms when you have two outcomes

Resist the urge to think that a situation with only two possible outcomes is

a 50-50 situation The only time a situation with two possible outcomes is a50-50 proposition is when both outcomes are equally likely to occur, as in theflip of a fair coin

I often ask students to tell me what they think the probability is that a ketball player will make a free throw Most students tell me the probabilitydepends on the player and his or her free-throw percentage (number of madeshots divided by the number of attempts) For example, basketball professionalShaquille O’Neal’s career best is 62 percent, shot in the 2002-2003 season

bas-When Shaq stepped up to the line that season, he made his free throws 62 cent of the time, and he missed them 38 percent of the time At any particularmoment during that season when he was standing at the line to make a freethrow, the chance of him making it was 62 percent However, a few students look

per-at me and say, “Wait a minute He either makes it or he doesn’t So, shouldn’this chance be 50-50?”

If you look at it from a strictly basketball point of view, that reasoning doesn’tmake sense, because everyone would be a 50-percent free throw shooter — nomore, no less — including people who don’t even play basketball! The proba-bility of making a free throw on your next try is based on a relative frequencyapproach (see the section “Find relative frequencies” earlier in this chapter) —

it depends on what percentage you’ve made over the long haul, and thatdepends on many factors, not chance alone

However, if you look at the situation from a probability point of view, it may behard to escape this misconception After all, you have two outcomes: make it

or miss it If you flip a coin, the probability of getting a head is 50 percent, andthe probability of getting a tail is 50 percent, so why doesn’t this hold true forfree throws? Because free throws aren’t set up like a fair coin Fair coins areequally likely to turn up heads or tails, and unless your free-throw percentage

is exactly 50 percent, you don’t shoot free throws like you toss coins

17

Chapter 1: The Probability in Everyday Life

Thinking that patterns can’t occur

What you perceive as random and what’s actually random are two differentthings Be careful not to misinterpret outcomes by identifying them as beingless probable because they don’t look random enough In other words, don’trule out the fact that patterns can and do occur over the long term, just bychance

The most important idea here is to not let your intuition get in the way ofreality Here are two examples to help you recognize what’s real and what’snot when it comes to probability

Picking a number from one to ten

Suppose that you ask a group of 100 people to pick a number from one to ten.(Go ahead and pick a number before reading on, just for fun.) You should expectabout ten people to pick one, ten people to pick two, and so on (not exactly,but fairly close) What happens, however, is that more people pick eitherthree or seven than the other numbers (Did you?) Why is this so? Becausemost people don’t want to pick one or ten because these numbers are on theends, and they don’t want to pick five because it rests in the middle, so they

go for numbers that appear more random — the middle of the numbers from

one to five (which is three) and the middle of the numbers from five to ten(which is seven) So, you throw the assumption that all ten numbers are equallylikely for selection out the window because people don’t think as objectively

as real random numbers do!

Research has shown that people can’t be objective enough in choosing randomnumbers, so to be sure that your probabilities can be repeated, you need tomake sure that you base them on random processes where each individualoutcome has an equal chance of selection If you put the numbers in a hat,shake, and pull one out, you create a random process

Flipping a coin ten times

Suppose that you flip a coin ten times and get the following result: H, T, H, T,

T, T, T, T, T, H People who see your recorded outcome may think that you made

up the results, because “you just don’t get six tails in a row.” Observers maythink your outcome just doesn’t look random enough Their intuition fuelstheir doubts, but their intuition is wrong In fact, you’re very likely to have

runs of heads or tails amongst a data set.

If you flip a coin ten times, with two possible outcomes on each flip, you have

2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1,024 possible outcomes, each one beingequally likely Your outcome with the coin is just as likely as one that maylook to be more random: H, T, H, T, H, T, H, T, H, T

18 Part I: The Certainty of Uncertainty: Probability Basics