Probabilities

The Little Numbers That Rule Our Lives

Peter Olofsson

• Publisher: Wiley
• Year: 2014
• ISBN:
  9781118898932
• Edition: 2
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Praise for the First Edition "If there is anything you want to know, or remind yourself, about probabilities, then look no further than this comprehensive, yet wittily written and enjoyable, compendium of how to apply probability calculations in real-world situations." - Keith Devlin, Stanford University, National Public Radio's "Math Guy" and author of The Math Gene and The Unfinished Game From probable improbabilities to regular irregularities, Probabilities: The Little Numbers That Rule Our Lives, Second Edition investigates the often surprising effects of risk and chance in our lives. Featuring a timely update, the Second Edition continues to be the go-to guidebook for an entertaining presentation on the mathematics of chance and uncertainty. The new edition develops the fundamental mathematics of probability in a unique, clear, and informal way so readers with various levels of experience with probability can understand the little numbers found in everyday life. Illustrating the concepts of probability through relevant and engaging real-world applications, the Second Edition features numerous examples on weather forecasts, DNA evidence, games and gambling, and medical testing. The revised edition also includes:

• The application of probability in finance, such as option pricing
• The introduction of branching processes and the extinction of family names
• An extended discussion on opinion polls and Nate Silver's election predictions

Probabilities: The Little Numbers That Rule Our Lives, Second Edition is an ideal reference for anyone who would like to obtain a better understanding of the mathematics of chance, as well as a useful supplementary textbook for students in any course dealing with probability.

Chapter 1
Computing Probabilities: Right Ways and Wrong Ways

The Probabilist

Whether you like it or not, probabilities rule your life. If you have ever tried to make a living as a gambler, you are painfully aware of this, but even those of us with more mundane life stories are constantly affected by these little numbers. Some examples from daily life where probability calculations are involved are the determination of insurance premiums, the introduction of new medications on the market, opinion polls, weather forecasts, and DNA evidence in courts. Probabilities also rule who you are. Did daddy pass you the X or the Y chromosome? Did you inherit grandma's big nose? And on a more profound level, quantum physicists teach us that everything is governed by the laws of probability. They toss around terms like the Schrödinger wave equation and Heisenberg's uncertainty principle, which are much too difficult for most of us to understand, but one thing they do mean is that the fundamental laws of physics can only be stated in terms of probabilities. And the fact that Newton's deterministic laws of physics are still useful can also be attributed to results from the theory of probabilities. Meanwhile, in everyday life, many of us use probabilities in our language and say things like “I'm 99% certain” or “There is a one-in-a-million chance” or, when something unusual happens, ask the rhetorical question “What are the odds?”

Some of us make a living from probabilities, by developing new theory and finding new applications, by teaching others how to use them, and occasionally by writing books about them. We call ourselves probabilists. In universities, you find us in mathematics and statistics departments; there are no departments of probability. The terms “mathematician” and “statistician” are much more well known than “probabilist,” and we are a little bit of both but we don't always like to admit it. If I introduce myself as a mathematician at a cocktail party, people wish they could walk away. If I introduce myself as a statistician, they do. If I introduce myself as a probabilist

well, most actually still walk away. They get upset that somebody who sounds like the Swedish Chef from the Muppet Show tries to impress them with difficult words. But some stay and give me the opportunity to tell them some of the things I will now tell you about.

Let us be etymologists for a while and start with the word itself, probability. The Latin roots are probare, which means to test, prove, or approve, and habilis, which means apt, skillful, able. The word “probable” was originally used in the sense “worthy of approval,” and its connection to randomness came later when it came to mean “likely” or “reasonable.” In my native Swedish, the word for probable is “sannolik,” which literally means “truthlike” as does the German word “wahrscheinlich.” The word “probability” still has room for nuances in the English language, and Merriam-Webster's online dictionary lists four slightly different meanings. To us a probability is a number used to describe how likely something is to occur, and probability (without the indefinite article) is the study of probabilities.

Probabilities are used in situations that involve randomness. Many clever people have thought about and debated what randomness really is, and we could get into a long philosophical discussion that could fill the rest of the book. Let's not. The French mathematician Pierre-Simon Laplace (1749–1827) put it nicely: “Probability is composed partly of our ignorance, partly of our knowledge.” Inspired by Monsieur Laplace, let us agree that you can use probabilities whenever you are faced with uncertainty. You could:

• Toss a coin, roll a die, or spin a roulette wheel
• Watch the stock market, the weather, or the Super Bowl
• Wonder if there is an oil well in your backyard, if there is life on Mars, if Elvis is alive

These examples differ from each other. The first three are cases where the outcomes are equally likely. Each individual outcome has a probability that is simply one divided by the number of outcomes. The probability is 1/2 to toss heads, 1/6 to roll a 6, and 1/38 to get the number 29 in roulette (an American roulette wheel has the numbers 1–36, 0, and 00). Pure and simple. We can also compute probabilities of groups of outcomes. For example, what is the probability to get an odd number when rolling a die? As there are three odd outcomes out of six total, ...