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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2018
Packed with practical tips and techniques for solving probability problems Increase your chances of acing that probability exam -- or winning at the casino! Whether you're hitting the books for a probability or statistics course or hitting the tables at a casino, working out probabilities can be problematic. This book helps you even the odds. Using easy-to-understand explanations and examples, it demystifies probability -- and even offers savvy tips to boost your chances of gambling success! Discover how to * Conquer combinations and permutations * Understand probability models from binomial to exponential * Make good decisions using probability * Play the odds in poker, roulette, and other games
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Probability For Dummies®
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Library of Congress Control Number: 2005938252
ISBN-13: 978-0-471-75141-0
ISBN-10: 0-471-75141-3
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 of Statistics at Kansas State University, where she won the distinguished Presidential Teaching Award and earned tenure and promotion in 1998. In 2000, she returned to Ohio State and is now a Statistics Education Specialist/Auxiliary Faculty Member for the Department of Statistics. Dr. Rumsey has served on the American Statistical Association’s Statistics Education 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 the subject of Statistics Education. Her particular research interests are curriculum materials development, teacher training and support, and immersive learning 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).
Dedication
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 this book; to Chrissy Guthrie for her continued excellence and for being a wonderful source of support as my project editor; and to Dr. Marjorie Bond, Monmouth College, for another invaluable technical review. Thanks to Josh Dials for his editing that kept things light. Thanks to Kythrie Silva for believing 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 to thank all the students I have had the privilege of teaching; you are the inspiration for all of my work.
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Probability For Dummies®
To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Probability For Dummies Cheat Sheet” in the Search box.
Table of Contents
Cover
About the Author
Introduction
About This Book
Conventions Used in This Book
What You’re Not to Read
Foolish Assumptions
How This Book Is Organized
Icons Used in This Book
Where to Go from Here
Part I: 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
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
Part II: 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
Chapter 5: Applying Counting Rules with Combinations and Permutations
Counting on Permutations
Counting Combinations
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 III: 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 (cdf)
Expected Value, Variance, and Standard Deviation of a Discrete Random Variable
Outlining the Discrete Uniform Distribution
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
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
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
Chapter 11: Sampling Distributions and the Central Limit Theorem
Surveying a Sampling Distribution
Gaining Access to Your Statistics through the Central Limit Theorem (CLT)
The Sampling Distribution of the Sample Total (t)
The Sampling Distribution of the Sample Mean,
The Sampling Distribution of the Sample Proportion,
Chapter 12: Investigating and Making Decisions with Probability
Confidence Intervals and Probability
Probability and Hypothesis Testing
Probability in Quality Control
Part IV: 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
Chapter 14: Covering All the Angles of the Geometric Distribution
Shaping Up the Geometric Distribution
Finding Probabilities for the Geometric by Using the pmf
Uncovering the Expected Value and Variance of the Geometric
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
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
Part V: 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
Drawing Up Probabilities for the Continuous Uniform Distribution
Corralling Cumulative Probabilities, Using F(x)
Figuring the Expected Value and Variance of the Continuous Uniform
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
Part VI: 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 (Plus One) Probability Mistakes
Forgetting 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 Your Reference
Binomial Table
Normal Table
Poisson Table
Connect with Dummies
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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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 could 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 — he 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.
About This Book
The main goal of this book is to cut down the amount of time you spend spinning your wheels to figure out a probability. The design of this book allows you to quickly find out how to solve the probability questions you’re asking (or that you have to answer).
This book gives you the tools to read, set up, and solve a wide range of probability problems. Because all probability problems tend to look different, I build strategies 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 the correct answer. You also gain practice interpreting probability and discovering what misconceptions and common errors you should avoid.
Along the way, you find some interesting surprises and a bird’s eye view of how probability pulls on the strings of the real world. I also include tips and strategies for playing games of chance, so if you do win the lottery, you can write 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 statistics courses, in addition to real-world probability topics. Most probability books out there help you win casino games but don’t help you out much with the probability problems you see in a probability and/or statistics course.
I provide an extensive number of examples to cover the many different types 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 a similar problem-solving strategy when you take exams.
My nonlinear approach allows you to skip around in the book and still have 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, and put into practice probability definitions, techniques, and processes.
I concentrate on clear and concise step-by-step procedures that intuitively explain how to work through probability problems and remember how to do them later on.
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 the Technical 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 contain fun and interesting, yet skippable, tidbits. I often use these sidebars to illustrate how people put probability to use in everyday life. Taking a moment to read the sidebars will enhance your understanding and appreciation of probability, but if you’re pressed for time or simply uninterested, you won’t miss out on any essential information.
Foolish Assumptions
I wrote this book for anyone who wants and/or needs to know about probability with little or no experience necessary. For students, you may be taking a course just in probability, and you’re 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 [Wiley], helps you with the statistics). But it 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.
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 references for you to use. Each part contains chapters that break down each major objective into understandable pieces.
Part I: The Certainty of Uncertainty: Probability Basics
This part gives you the fundamentals of probability, along with strageties for setting up and solving the most common probability problems in the introductory 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 goes against our intuition. You discover the basic definitions, terms, notation, and rules for probability, and you get answers to those all-important (and often frustrating) questions that perplex students of probability, such as, “What’s the 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 discover good strategies for solving more complex probability problems involving the Law of Total Probability and Bayes’ Theorem.
Part II: Counting on Probability and Betting to Win
In this part, you get down to the nitty gritty of probability, solving problems that involve 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 evaluating probability models. You discover all the ins and outs of a probability distribution; the basic concepts and rules for defining probability distributions; and how to find probabilities, means, and variances. You work with the binomial and normal distributions, and you find out how probability ties in to the major results from statistics: the Central Limit Theorem, hypothesis testing, and overall decision making in the real world.
Part IV: Taking It Up a Notch: Advanced Probability Models
In this part, you work with more intermediate probability models that count and try to predict the number of arrivals, successes, or the number of trials needed to achieve a certain goal. The probability distributions I focus on are the Poisson, negative binomial, geometric, and hypergeometric. You find out how 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 getting 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 statistics courses that have calculus as a prerequisite — mainly the uniform (continuous case) distribution, exponential distribution, and other userdefined probability density functions. You see how to find probabilities and the expected value, variance, and standard deviation of continuous probability 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 methods of solution that don’t use calculus for the uniform and exponential.
Part VI: The Part of Tens
In this part, you find my top tens lists: ten steps to a better probability grade and ten probability misconceptions and how to avoid them. This information is based on my years of experience teaching, answering questions, writing questions, and grading homework. This part will help you pinpoint the most important ideas in probability and the most common errors that are made. It also 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 reference. These tables help you find probabilities for the binomial distribution, the normal distribution, and the Poisson distribution.
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).
These I save for particular ideas that I hope you’ll remember long after you read this book. They mainly refer 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. The info 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!
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 misconceptions about probability that can get you into trouble.
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 to people who are unsure about where to start:
If you’re taking a probability or statistics class based in algebra, I recommend starting with
Part I
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 IV
and work your way to
Part V
. In
Part V
, 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 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 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 three green balls and four red balls from an urn that contains twenty green balls and 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 that happening?” Someone wins the lottery not once, but twice. You accidentally run into a friend you haven’t seen since high school during a vacation in Florida. 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 the random phenomena of life. But it also helps you realize the limitations of probability and why probabilities can take you only so far.
In this chapter, you observe the impact of probability on your everyday life and some of the ways people come up with probabilities. You also find out that 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 between zero and one — a proportion, in other words. You can write it as a percentage, because people like to talk about probability as a percentage chance, or you 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 probability of an event happening to the probability of the event not happening. For example, if the probability of a horse winning a race is 50 percent (), the odds of this horse winning are . So the odds are 1 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 a chance 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 around the 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 for which the outcomes are never certain), you know that certain outcomes can happen, 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 to occur in the long term (or over many individuals)?
If the chance of rain tomorrow is 30 percent, does that mean it won’t rain because 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 the other 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 the previous section), it may be easier to see how they apply to a group rather than to an individual. But sometimes one way makes more sense than the other, depending on the situation you face. The following sections outline ways to interpret probabilities as they apply to groups or individuals so you don’t run into misinterpretation problems.
Playing the instant lottery
Probabilities are based on long-term percentages (over thousands of trials), so when you apply them to a group, the group has to be large enough (the larger the better, but at least 1,500 or so items or individuals) for the probabilities to really apply. Here’s an example where long-term interpretation makes sense in place of short-term interpretation. Suppose the chance of winning a prize in an instant lottery game is , or 10 percent. This probability means that in the long term (over thousands of tickets), 10 percent of all instant lottery tickets purchased for this game will win a prize, and 90 percent won’t. It doesn’t mean that if you buy 10 tickets, one of them will automatically win.
If you buy many sets of 10 tickets, on average, 10 percent of your tickets will win, 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, you might expect a high chance of winning at least one prize. But the chance of you 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 probability with 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 community are Democrats, 30 percent are Republicans, and the remaining 10 percent are Independents or have another political affiliation. If you randomly select one person from your community, what’s the chance the person is a Democrat? The chance is 60 percent. You can’t say that the person is surely a Democrat because the chance is over 50 percent; the percentages just tell you that the person 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 certain genetic disorders. Before you sign the papers to have surgery, doctors and nurses tell you about the chances that you’ll have complications. And before you buy a vehicle, you can find out probabilities for almost every topic regarding that vehicle, including the chance of repairs becoming necessary, of the vehicle 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 fact based on probability).
While scanning the Internet, I quickly found several examples of probabilities that 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 medications properly is higher if companies put the medications in the new packaging than it is when the companies put the medicines in bottles. You don’t know what 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.
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 study recorded the number of cars stolen each year in various metropolitan areas of Ohio. Note that the study reports the information as the number of thefts per 100,000 vehicles. The researchers needed a fixed number of vehicles in order to be fair about the comparison. If the study used only the number of thefts, cities with more cars would always rank higher than cities with fewer cars.
How did the researchers get the specific numbers for this study? They took the actual number of thefts and divided it by the total number of vehicles 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 the rates as probabilities, they simply divided them by 100,000 to put them back in decimal form. For Toledo, the probability of car theft is , or 0.58 percent; for Columbus, the probability 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 a tropical storm developing into a hurricane that will ultimately make landfall at a certain place and time — a probability that depends on many elements that are themselves nearly impossible to determine. If people calculate actual probabilities for hurricane outcomes, they make estimates at best.
Some probabilities, on the other hand, are very easy to calculate for an exact number, 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 probability 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.
After you analyze the complexity of the situation, you can use one of four major approaches 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 scientific. It’s based mostly on opinions, feelings, or hopes, meaning that you typically don’t use this type of probability approach in real scientific endeavors. You basically say, “Here’s what I think the probability is.” For example, although the actual, true probability that the Ohio State football team will win the national championship is out there somewhere, no one knows what it is, even though 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 hate Ohio 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 this approach isn’t scientific, it sure makes for some great sports talk amongst the fans!
Take a classical approach
The classical approach to probability is a mathematical, formula-based approach. You can use math and counting rules to calculate exact probabilities in many cases (for more on the counting rules, see Chapter 5). Anytime you have a situation where you can enumerate the possible outcomes and figure their individual probabilities by using math, you can use the classical approach to getting the probability of an outcome or series of outcomes from a random process.
For example, when you roll two die, you have six possible outcomes for the first die, and for each of those outcomes, you have another six possible outcomes for the second die. All together, you have possible outcomes for the pair. In order to get a sum of two on a roll, you have to roll two 1s, meaning it can happen in only one way. So, the probability of getting a sum of two is . The probability of getting a sum of three is , because only two of the outcomes result in a sum of three: 1-2 or 2-1. A sum of seven has a probability of , or — the highest probability of any sum of two die. Why is seven the sum with highest 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 the gambling game craps. (For more on this example, see Chapter 2.)
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 the probability of achieving success when you’re trying to make a sale is the same on each trial, you can use the binomial probability model for figuring out the probability of making 5 sales in 20 tries. Many types of probability models are available, and I discuss many of them in this book. (For more on the binomial probability model, see Chapter 8.)
The classical approach doesn’t work when you can’t describe the possible individual outcomes and come up with some mathematical way of determining the probabilities. For example, if you have to decide between different brands 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 help you for a couple reasons. First, you can’t assume that the probability of a refrigerator 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 to figure out the chances of repairs for different brands of refrigerators; it depends 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 to figure a probability, the relative frequency approach is your best bet. The approach 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 probabilities for the refrigerator repairs example in the previous section with the relative frequency approach by collecting data on refrigerator repair records.)
Suppose, for example, that you’re watching your birdfeeder, and you notice a lot of cardinals coming for dinner. You want to find the probability that the next bird 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 of time 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 you observe, 27 out of 100 visits — or 27 percent, the relative frequency — were made 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.
Consuming data with Consumer Reports
The magazine Consumer Reports — put out by the Consumers Union, a nonprofit group that helps provide consumer protection information — does thousands of studies to test different makes and models 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, the group comes up with a list of recommendations regarding 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 various models for energy efficiency, temperature performance, noise, ease of use, and energy cost per year. The researchers figure out what percentage of time the refrigerators need repairs, don’t perform properly, and so on, and they base their reports on what they find.
A limitation of the relative frequency approach is that the probabilities you come up with are only estimates because you base them on finite samples of data you collect. And those estimates are only as good as the data that you collect. For example, if you collected your birdfeeder data when you offered sunflower seeds, but now you offer thistle seed (loved by smaller birds), your probability 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 the finches are also out and about. The issue of collecting good data is a statistical one; see Statistics For Dummies (Wiley) for more information.
Use simulations
The simulation approach is a process that creates data by setting up a certain scenario, playing out that scenario over and over many times, and looking at the percentage of times a certain outcome occurs. It may sound like the relative frequency approach (see the previous section), but it’s different in three ways:
You create the data (usually with a computer); you don’t collect it out in the real world.
The amount of data is typically much larger than the amount you could observe in real life.
You use a certain model that scientists come up with, and models have assumptions.
Tracking down hurricanes
One major area where professionals use computer models is in predicting the arrival, intensity, and path of tropical storms, including hurricanes. Computer hurricane models help scientists and leaders perform integrated cost-benefit studies; evaluate the effects of regulatory policies; and make decisions during crises. Insurance companies use the models to make predictions regarding the number of and estimated damage due to future hurricanes, which helps them adjust their premiums appropriately to be ready to pay out the huge claims that come with large hurricanes.
Computer models for tropical storms are best at predicting long-run (versus short-term) losses across large (versus small) geographic areas, due to the high margin of error. Margin of error is the amount by which your results are expected to change from sample to sample. You can’t look at a single storm and say exactly what’s going to happen. AIR Worldwide, whose computer models are used by half the residential property insurance markets in Florida and 85 percent of the companies that underwrite insurers, calculates projections over storms across a 50,000-year span. Another modeling expert recently lengthened its computer modeling from 100,000 to 300,000 years to get results within an acceptable margin of error.
The models contain so many variables that it takes many trials to approach a predictable average. Flipping a coin, for instance, has only one variable with two outcomes. If you want to estimate the probability of flipping heads by using a model, it takes about 2,500 trials to get a result within a 2-percent margin of error. The more variables, the more trials required to get a dependable outcome. And with hurricanes, the number of variables is huge. The computer models used by the National Hurricane Center include variables such as the initial latitude and longitude of the storm, the components of the “storm motion vector,” and the initial storm intensity, just to name a few.
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 thousands of times and see what you end up with. Change the probabilities of heads and tails to see what happens. Your experiments are examples of simple simulations.
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 remember very clearly a simulation that a student performed to predict which team would win the NCAA basketball tournament some years ago. The student gave each of the 64 teams in the tournament a probability of winning its game based on certain statistics that the sports gurus came up with. The student fed those probabilities into the computer and made the computer repeat the tournament over and over millions of times, recording who won each game and who won the entire tournament. On 96 percent of the simulations, Duke University won the whole thing. So, of course, it seemed as if Duke was a shoe-in that year. Guess how long Duke actually lasted? The team went down in the second of six rounds.
Probability Misconceptions to Avoid
No matter how researchers calculate a probability or what kind of information or data they base it on, the probability is often misinterpreted or applied in the wrong way by the media, the public, and even other researchers who don’t quite understand the limitations of probability. The main idea is that probability often goes against your intuition, and you have to be very careful about not letting your intuition get the better of you when thinking in terms of probability. This section highlights some of the most common misconceptions about probability.
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 a 50-50 proposition is when both outcomes are equally likely to occur, as in the flip of a fair coin.
I often ask students to tell me what they think the probability is that a basketball player will make a free throw. Most students tell me the probability depends on the player and his or her free-throw percentage (number of made shots divided by the number of attempts). For example, basketball professional Shaquille O’Neal’s career best is 62 percent, shot in the 2002-2003 season. When Shaq stepped up to the line that season, he made his free throws 62 percent of the time, and he missed them 38 percent of the time. At any particular moment during that season when he was standing at the line to make a free throw, the chance of him making it was 62 percent. However, a few students look at me and say, “Wait a minute. He either makes it or he doesn’t. So, shouldn’t his chance be 50-50?”
If you look at it from a strictly basketball point of view, that reasoning doesn’t make sense, because everyone would be a 50-percent free throw shooter — no more, no less — including people who don’t even play basketball! The probability of making a free throw on your next try is based on a relative frequency approach (see the section “Find relative frequencies” earlier in this chapter) — it depends on what percentage you’ve made over the long haul, and that depends on many factors, not chance alone.
However, if you look at the situation from a probability point of view, it may be hard 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, and the probability of getting a tail is 50 percent, so why doesn’t this hold true for free throws? Because free throws aren’t set up like a fair coin. Fair coins are equally 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.
Thinking that patterns can’t occur
What you perceive as random and what’s actually random are two different things. Be careful not to misinterpret outcomes by identifying them as being less probable because they don’t look random enough. In other words, don’t rule out the fact that patterns can and do occur over the long term, just by chance.
The most important idea here is to not let your intuition get in the way of reality. Here are two examples to help you recognize what’s real and what’s not 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 expect about 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 either three or seven than the other numbers. (Did you?) Why is this so? Because most people don’t want to pick one or ten because these numbers are on the ends, 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 equally likely 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 random numbers, so to be sure that your probabilities can be repeated, you need to make sure that you base them on random processes where each individual outcome 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 may think your outcome just doesn’t look random enough. Their intuition fuels their 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 possible outcomes, each one being equally likely. Your outcome with the coin is just as likely as one that may look to be more random: H, T, H, T, H, T, H, T, H, T.
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
Exploring the difference between independence and exclusivity
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 the 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. Rolling a single die is an example of a finite random process because you can achieve only six possible outcomes, and you can account for them all. Probability models that you can use for finite sample spaces include the binomial (see Chapter 8), the discrete uniform (see Chapter 7), and the hypergeometric (see Chapter 16).
Countably infinite sample spaces
Countably infinite means that you have a way to show the progression of the values, but they can go on into infinity. For example, if your random process involves the number of phone calls that come in to a switchboard during a week’s time, the possible outcomes of S aren’t finite, but rather countably infinite. In this case, . S goes to infinity because you can’t be sure of the maximum number of calls coming in. If you count all the calls, you get a fixed number with a countably infinite sample space S, but to be sure you allow for any maximum, you let S go on to infinity.
The way to get around the strange countably infinite situation is to give progressively smaller and smaller probabilities to the larger and larger values of S so eventually they become irrelevant. (More on this probability model in Chapter 13.)
Uncountably infinite sample spaces
Uncountably infinite
