Probability Demystified - 2005

Probability Demystified - 2005

(Parte 1 de 4)

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Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

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To all of my teachers, whose examples instilled in me my love of mathematics and teaching.

Preface ix Acknowledgments xi

CHAPTER 1 Basic Concepts 1 CHAPTER 2 Sample Spaces 2 CHAPTER 3 The Addition Rules 43 CHAPTER 4 The Multiplication Rules 56 CHAPTER 5 Odds and Expectation 7 CHAPTER 6 The Counting Rules 94 CHAPTER 7 The Binomial Distribution 114 CHAPTER 8 Other Probability Distributions 131 CHAPTER 9 The Normal Distribution 147 CHAPTER 10 Simulation 177 CHAPTER 1 Game Theory 187 CHAPTER 12 Actuarial Science 210

Final Exam 229 Answers to Quizzes and Final Exam 244 Appendix: Bayes’ Theorem 249 Index 255 vii

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‘‘The probable is what usually happens.’’ — Aristotle

Probability can be called the mathematics of chance. The theory of probability is unusual in the sense that we cannot predict with certainty the individual outcome of a chance process such as flipping a coin or rolling a die (singular for dice), but we can assign a number that corresponds to the probability of getting a particular outcome. For example, the probability of getting a head when a coin is tossed is 1/2 and the probability of getting a two when a single fair die is rolled is 1/6.

We can also predict with a certain amount of accuracy that when a coin is tossed a large number of times, the ratio of the number of heads to the total number of times the coin is tossed will be close to 1/2.

Probability theory is, of course, used in gambling. Actually, mathematicians began studying probability as a means to answer questions about gambling games. Besides gambling, probability theory is used in many other areas such as insurance, investing, weather forecasting, genetics, and medicine, and in everyday life.

What is this book about? First let me tell you what this book is not about:

. This book is not a rigorous theoretical deductive mathematical approach to the concepts of probability.

. This book is not a book on how to gamble. And most important ix Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.

. This book is not a book on how to win at gambling!

This book presents the basic concepts of probability in a simple, straightforward, easy-to-understand way. It does require, however, a knowledge of arithmetic (fractions, decimals, and percents) and a knowledge of basic algebra (formulas, exponents, order of operations, etc.). If you need a review of these concepts, you can consult another of my books in this series entitled Pre-Algebra Demystified.

This book can be used to gain a knowledge of the basic concepts of probability theory, either as a self-study guide or as a supplementary textbook for those who are taking a course in probability or a course in statistics that has a section on probability.

The basic concepts of probability are explained in the first two chapters.

Then the addition and multiplication rules are explained. Following that, the concepts of odds and expectation are explained. The counting rules are explained in Chapter 6, and they are needed for the binomial and other probability distributions found in Chapters 7 and 8. The relationship between probability and the normal distribution is presented in Chapter 9. Finally, a recent development, the Monte Carlo method of simulation, is explained in Chapter 10. Chapter 1 explains how probability can be used in game theory and Chapter 12 explains how probability is used in actuarial science. Special material on Bayes’ Theorem is presented in the Appendix because this concept is somewhat more difficult than the other concepts presented in this book.

In addition to addressing the concepts of probability, each chapter ends with what is called a ‘‘Probability Sidelight.’’ These sections cover some of the historical aspects of the development of probability theory or some commentary on how probability theory is used in gambling and everyday life.

I have spent my entire career teaching mathematics at a level that most students can understand and appreciate. I have written this book with the same objective in mind. Mathematical precision, in some cases, has been sacrificed in the interest of presenting probability theory in a simplified way. Good luck!

Allan G. Bluman

PREFACEx

I would like to thank my wife, Betty Claire, for helping me with the preparation of this book and my editor, Judy Bass, for her assistance in its publication. I would also like to thank Carrie Green for her error checking and helpful suggestions.

xi Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.

CHAPTER 1

Basic Concepts

Introduction

Probability can be defined as the mathematics of chance. Most people are familiar with some aspects of probability by observing or playing gambling games such as lotteries, slot machines, black jack, or roulette. However, probability theory is used in many other areas such as business, insurance, weather forecasting, and in everyday life.

In this chapter, you will learn about the basic concepts of probability using various devices such as coins, cards, and dice. These devices are not used as examples in order to make you an astute gambler, but they are used because they will help you understand the concepts of probability.

1 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.

Probability Experiments

Chance processes, such as flipping a coin, rolling a die (singular for dice), or drawing a card at random from a well-shuffled deck are called probability experiments.A probability experiment is a chance process that leads to welldefined outcomes or results. For example, tossing a coin can be considered a probability experiment since there are two well-defined outcomes—heads and tails.

An outcome of a probability experiment is the result of a single trial of a probability experiment. A trial means flipping a coin once, or drawing a single card from a deck. A trial could also mean rolling two dice at once, tossing three coins at once, or drawing five cards from a deck at once. A single trial of a probability experiment means to perform the experiment one time.

The set of all outcomes of a probability experiment is called a sample space. Some sample spaces for various probability experiments are shown here.

Experiment Sample Space Toss one coin H, T* Roll a die 1, 2, 3, 4, 5, 6

Toss two coins H, HT, TH, T *H=heads; T=tails.

Notice that when two coins are tossed, there are four outcomes, not three.

Consider tossing a nickel and a dime at the same time. Both coins could fall heads up. Both coins could fall tails up. The nickel could fall heads up and the dime could fall tails up, or the nickel could fall tails up and the dime could fall heads up. The situation is the same even if the coins are indistinguishable.

It should be mentioned that each outcome of a probability experiment occurs at random. This means you cannot predict with certainty which outcome will occur when the experiment is conducted. Also, each outcome of the experiment is equally likely unless otherwise stated. That means that each outcome has the same probability of occurring.

When finding probabilities, it is often necessary to consider several outcomes of the experiment. For example, when a single die is rolled, you may want to consider obtaining an even number; that is, a two, four, or six. This is called an event. An event then usually consists of one or more

CHAPTER 1 Basic Concepts2 outcomes of the sample space. (Note: It is sometimes necessary to consider an event which has no outcomes. This will be explained later.)

An event with one outcome is called a simple event. For example, a die is rolled and the event of getting a four is a simple event since there is only one way to get a four. When an event consists of two or more outcomes, it is called a compound event. For example, if a die is rolled and the event is getting an odd number, the event is a compound event since there are three ways to get an odd number, namely, 1, 3, or 5.

Simple and compound events should not be confused with the number of times the experiment is repeated. For example, if two coins are tossed, the event of getting two heads is a simple event since there is only one way to get two heads, whereas the event of getting a head and a tail in either order is a compound event since it consists of two outcomes, namely head, tail and tail, head.

EXAMPLE: A single die is rolled. List the outcomes in each event:

a. Getting an odd number b. Getting a number greater than four c. Getting less than one a. The event contains the outcomes 1, 3, and 5. b. The event contains the outcomes 5 and 6.

c. When you roll a die, you cannot get a number less than one; hence, the event contains no outcomes.

Classical Probability

Sample spaces are used in classical probability to determine the numerical probability that an event will occur. The formula for determining the probability of an event E is

PðEÞ¼ number of outcomes contained in the event E total number of outcomes in the sample space

CHAPTER 1 Basic Concepts 3

EXAMPLE: Two coins are tossed; find the probability that both coins land heads up.

The sample space for tossing two coins is H, HT, TH, and T. Since there are 4 events in the sample space, and only one way to get two heads (H), the answer is

EXAMPLE: A die is tossed; find the probability of each event:

a. Getting a two b. Getting an even number c. Getting a number less than 5

The sample space is 1, 2, 3, 4, 5, 6, so there are six outcomes in the sample space.

a. P(2) ¼ 1 6 , since there is only one way to obtain a 2.

b. P(even number) ¼ 36 ¼ 1 2 , since there are three ways to get an odd c. P(number less than 5Þ¼ 46 ¼ 2 3 , since there are four numbers in the sample space less than 5.

EXAMPLE: A dish contains 8 red jellybeans, 5 yellow jellybeans, 3 black jellybeans, and 4 pink jellybeans. If a jellybean is selected at random, find the probability that it is a. A red jellybean b. A black or pink jellybean c. Not yellow d. An orange jellybean

CHAPTER 1 Basic Concepts4

d. P(orange)= 020 ¼ 0, since there are no orange jellybeans.

Probabilities can be expressed as reduced fractions, decimals, or percents.

For example, if a coin is tossed, the probability of getting heads up is 12 or 0.5 or 50%. (Note: Some mathematicians feel that probabilities should be expressed only as fractions or decimals. However, probabilities are often given as percents in everyday life. For example, one often hears, ‘‘There is a 50% chance that it will rain tomorrow.’’)

Probability problems use a certain language. For example, suppose a die is tossed. An event that is specified as ‘‘getting at least a 3’’ means getting a 3, 4, 5, or 6. An event that is specified as ‘‘getting at most a 3’’ means getting a1 ,2 ,o r3 .

Probability Rules

There are certain rules that apply to classical probability theory. They are presented next.

(Parte 1 de 4)

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