Probability For Dummies (eBook)

Deborah J. Rumsey (Autor)

eBook Download: EPUB

2024 | 2. Auflage
400 Seiten
For Dummies (Verlag)
978-1-394-28189-3 (ISBN)

Infos

Learn how to calculate your chances with easy-to-understand explanations of probability

Probability-the likelihood or chance of an event occurring-is an important branch of mathematics used in business and economics, finance, engineering, physics, and beyond. We see probability at work every day in areas such as weather forecasting, investing, and sports betting. Packed with real-life examples and mathematical problems with thorough explanations, Probability For Dummies helps students, professionals, and the everyday reader learn the basics. Topics include set theory, counting, permutations and combinations, random variables, conditional probability, joint distributions, conditional expectations, and probability modeling. Pass your probability class and play your cards right, with this accessible Dummies guide.

• Understand how probability impacts daily life
• Discover what counting rules are and how to use them
• Practice probability concepts with sample problems and explanations
• Get clear explanations of all the topics in your probability or statistics class

Probability For Dummies is the perfect Dummies guide for college students, amateur and professional gamblers, investors, insurance professionals, and anyone preparing for the actuarial exam.

Deborah J. Rumsey, PhD, is a Professor of Teaching Practice at The Ohio State University. Dr. Rumsey is a Fellow of the American Statistical Association, and the author of several For Dummies titles, including Statistics For Dummies and Statistics Workbook For Dummies.

Textauszug

Chapter 1

The Probability in Everyday Life

IN THIS CHAPTER

Recognizing the prevalence and impact of probability in your daily 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 while you’re on vacation in Italy. A cop pulls you over the one time you forget to put your seat belt 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. There are many types of odds, and each has its own calculation; one popular set of odds is the odds for an event. This 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 rolling a 1 on a die is ⅙, the odds for rolling a 1 on a die are ⅙ divided by ⅚ (⅚ being the probability of not getting a 1), which is 1 to 5. Conversely, the odds against an event are found by taking the probability of the event not happening divided by the probability of the event happening. For example, the odds against getting a 1 on a die are ⅚ divided by ⅙ or 5 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 (0.8), 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 of 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 preceding 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 ten tickets, one of them will automatically win.

If you buy many sets of ten tickets, on average, 10 percent of your tickets will win, but sometimes a group of ten has multiple winners, and sometimes it has no winners. The winners are mixed up amongst the total population of tickets. If you buy exactly ten tickets, each with a 10 percent chance of winning, you may expect a high chance of winning at least one prize. But the chance of you winning at least one prize with those ten 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, they’re either a Democrat or not — they can’t be 60 percent Democrat.

Seeing probability in everyday life

Probabilities affect the biggest and smallest decisions of people’s lives. For example, pregnant women look at the probabilities of their babies having certain genetic disorders. Or before you sign the papers to have surgery, doctors and nurses tell you about the chances that you’ll have complications. 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 your surviving a front-end crash or rollover (the latter depends on whether you wear a seat belt — another fact based on probability).

While scanning the internet, I quickly found several examples of probabilities that affect people’s everyday lives. Here are two of them:

• According to the Colorado State University Tropical Weather and Climate Research Center, 2024 was expected to have a higher average hurricane activity compared to the seasons between 1991 and 2020, as reported by the National Association of Home Builders. The probability of major hurricanes making landfall in 2024 was:

  • Sixty-two percent for the entire U.S. coastline (average from 1880–2020 is 43 percent)
  • Thirty-four percent for the U.S. East Coast, including the Florida peninsula (average from 1880–2020 is 21 percent)
  • Forty-two percent for the Gulf Coast from the Florida panhandle westward to Brownsville (average from 1880–2020 is 27 percent)
  • Sixty-six percent for the Caribbean (average from 1880–2020 is 47 percent)

  One of the ways researchers develop probabilities for events such as hurricanes, is through computer simulations and modeling. This process sets up a model that contains events that occur with certain probabilities based on prior research and data collection, and it runs the model over and over again, recording the outcomes each time. So, for example, if the probability that a hurricane will hit the U.S. coastline is 62 percent, that means the model had this outcome 62 percent of the times it was repeated.

• According to the National Insurance Crime Bureau, the top three cities for auto theft in the United States as of this writing are Bakersfield, California (with 1,023 thefts per 100,000 cars); Denver, Colorado (with 964 thefts per 100,000 cars); and Pueblo, Colorado (with 891 thefts per 100,000 cars).

  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 the United States. 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.

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Produkt-Details

• Erscheint lt. Verlag: 14.10.2024
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Schlagworte: Applied mathematics • basic probability • Basic Statistics • counting rules • Gambling • gambling dummies • intro to probability • intro to statistics • Odds • Probability • probability book • probability modeling • Probability Theory • set theory • Statistics
• ISBN-10: 1-394-28189-7 / 1394281897
• ISBN-13: 978-1-394-28189-3 / 9781394281893