# Introduction to Probability

## First EditionMark Ward; Ellen Gundlach

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*Introduction to Probability* reaches out to a much wider introductory-level audience. Its conversational style, highly visual approach, practical examples, and step-by-step problem solving procedures help all kinds of students understand the basics of probability theory and its broad applications. The book was extensively class-tested through its preliminary edition, to make it even more effective at building confidence in students who have viable problem-solving potential but are not fully comfortable in the culture of mathematics.

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Learn More## Table of Contents

**I Randomness **1 Outcomes, Events, and Sample Spaces

1.1 Introduction

1.2 Complements and DeMorgan's Laws

1.3 Exercises

1.3.1 Practice

1.3.2 Extensions

1.3.3 Advanced

**2 Probability**

2.1 Introduction

2.2 Equally-Likely Events

2.3 Complementary Probabilities; Probabilities of Subsets

2.4 Inclusion-Exclusion

2.5 More Examples of Probabilities of Events

2.6 Exercises

2.6.1 Practice

2.6.2 Extensions

2.6.3 Advanced

3 Independent Events

3 Independent Events

3.1 Introduction

3.2 Some Nice Facts About Independence

3.3 Probability of Good Occurring Before Bad

3.4 Exercises

3.4.1 Practice

3.4.2 Extensions

3.4.3 Advanced

**4 Conditional Probability**

4.1 Introduction

4.2 Distributive Laws

4.3 Conditional Probabilities Satisfy the Standard Probability Axioms

4.4 Exercises

4.4.1 Practice

4.4.2 Extensions

4.4.3 Advanced

**5 Bayes' Theorem**

5.1 Introduction to Versions of Bayes' Theorem

5.2 Multiplication with Conditional Probabilities

5.3 Exercises

5.3.1 Practice

5.3.2 Extensions

5.3.3 Advanced

**6 Review of Randomness**

6.1 Summary of Randomness

6.2 Exercises

**II Discrete Random Variables**

**7 Discrete Versus Continuous Random Variables**

7.1 Introduction

7.2 Examples

7.3 Exercises

7.3.1 Practice

7.3.2 Extensions

7.3.3 Advanced

**8 Probability Mass Functions and CDFs**

8.1 Introduction

8.2 Examples

8.3 Properties of the Mass and CDF

8.4 More Examples

8.5 Exercises

8.5.1 Practice

8.5.2 Extensions

8.5.3 Advanced

**9 Independence and Conditioning**

9.1 Joint Probability Mass Functions

9.2 Independent Random Variables

9.3 Three or More Random Variables That Are Independent

9.4 Conditional Probability Mass Functions

9.5 Exercises

9.5.1 Practice

9.5.2 Extensions

9.5.3 Advanced

**10 Expected Values of Discrete Random Variables**

10.1 Introduction

10.2 Examples

10.3 Exercises

10.3.1 Practice

10.3.2 Extensions

**11 Expected Values of Sums of Random Variables**

11.1 Introduction

11.2 Examples

11.3 Exercises

11.3.1 Practice

11.3.2 Extensions

**12 Variance of Discrete Random Variables**

12.1 Introduction.

12.2 Variance

12.3 Five Friendly Facts with Independence

12.4 Exercises

12.4.1 Practice

12.4.2 Extensions

12.4.3 Advanced

**13 Review of Discrete Random Variables**

13.1 Summary of Discrete Random Variables

13.2 Exercises

III Named Discrete Random Variables

III Named Discrete Random Variables

**14 Bernoulli Random Variables**

14.1 Introduction

14.2 Examples

14.3 Exercises

14.3.1 Practice

14.3.2 Extensions

14.3.3 Advanced

**15 Binomial Random Variables**

15.1 Introduction

15.2 Examples

15.3 Exercises

15.3.1 Practice

15.3.2 Extensions

15.3.3 Advanced

**16 Geometric Random Variables**

6.1 Introduction

16.2 Special Features of the Geometric Distribution

16.3 The Number of Failures

16.4 Geometric Memoryless Property

16.5 Random Variables That Are Not Geometric

16.6 Exercises

16.6.1 Practice

16.6.2 Extensions

16.6.3 Advanced

**17 Negative Binomial Random Variables**

17.1 Introduction

17.2 Examples

17.3 Exercises

17.3.1 Practice

17.3.2 Extensions

**18 Poisson Random Variables**

18.1 Introduction

18.2 Sums of Independent Poisson Random Variables

18.3 Using the Poisson as an Approximation to the Binomial

18.4 Exercises

18.4.1 Practice

18.4.2 Extensions

18.4.3 Advanced

**19 Hypergeometric Random Variables**

19.1 Introduction

19.2 Examples

19.3 Using the Binomial as an Approximation to the Hypergeometric

19.4 Exercises

19.4.1 Practice

19.4.2 Extensions

20 Discrete Uniform Random Variables

20 Discrete Uniform Random Variables

20.1 Introduction

20.2 Examples

20.3 Exercises

20.3.1 Practice

20.3.2 Extensions

20.3.3 Advanced

**21.1 Summing up: How do you tell all these random variables apart?**

21 Review of Named Discrete Random Variables

21 Review of Named Discrete Random Variables

21.2 Exercises

21.3 Review Problems

**IV Counting**

22 Introduction to Counting

22.1 Introduction

22 Introduction to Counting

22.2 Sampling With Versus Without Replacement; With Versus Without Regard to Order

22.3 Counting: Seating Arrangements

22.4 Exercises

22.4.1 Practice

22.4.2 Extensions

22.4.3 Seating Arrangement Problems

**23 Two Case Studies in Counting**

23.1 Poker Hands

23.1.1 Straight Flush

23.1.2 Four Of A Kind

23.1.3 Full House

23.1.4 Flush

23.1.5 Straight

23.1.6 Three Of A Kind

23.1.7 Two Pair

23.1.8 One Pair

23.2 Yahtzee

23.2.1 Upper Section

23.2.2 Three Of A Kind

23.2.3 Four Of A Kind

23.2.4 Full House

23.2.5 Small Straight

23.2.6 Large Straight

23.2.7 Yahtzee

**24.1 Introduction**

V Continuous Random Variables

24 Continuous Random Variables and PDFs

V Continuous Random Variables

24 Continuous Random Variables and PDFs

24.2 Examples

24.3 Exercises

24.3.1 Practice

24.3.2 Extensions

24.3.3 Advanced

**25 Joint Densities**

25.1 Introduction

25.2 Examples

25.3 Exercises

25.3.1 Practice

25.3.2 Extensions

25.3.3 Advanced

**26 Independent Continuous Random Variables**

26.1 Introduction

26.2 Examples

26.3 Exercises

26.3.1 Practice

26.3.2 Extensions

26.3.3 Advanced

27 Conditional Distributions

27 Conditional Distributions

27.1 Introduction

27.2 Examples

27.3 Exercises

27.3.1 Practice

27.3.2 Extensions

28 Expected Values of Continuous Random Variables

28 Expected Values of Continuous Random Variables

28.1 Introduction

28.2 Some Generalizations about Expected Values

28.3 Some Applied Problems with Expected Values

28.4 Exercises

28.4.1 Practice

28.4.2 Extensions

28.4.3 Advanced

**29.1 Variance of a Continuous Random Variable**

29 Variance of Continuous Random Variables

29 Variance of Continuous Random Variables

29.2 Expected Values of Functions of One Continuous Random Variable

29.3 Expected Values of Functions of Two Continuous Random Variables

29.4 More Friendly Facts about Continuous Random Variables

29.5 Exercises

29.5.1 Practice

29.5.2 Extensions

29.5.3 Advanced

**30 Review of Continuous Random Variables**

30.1 Summary of Continuous Random Variables

30.2 Exercises

**VI Named Continuous Random Variables 31 Continuous Uniform Random Variables**31.1 Introduction

31.2 Examples

31.3 Linear Scaling of a Uniform Random Variable

31.4 Exercises

31.4.1 Practice

31.4.2 Extensions

31.4.3 Advanced

**32 Exponential Random Variables**

32.1 Introduction

32.2 Average and Variance

32.3 Properties of Exponential Random Variables

32.3.1 Complement of the CDF

32.3.2 Memoryless Property of Exponential Random Variables

32.3.3 Minimum of Independent Exponential Random Variables

32.3.4 Poisson Process

32.3.5 Moments of an Exponential Random Variable (Optional)

32.4 Exercises .

32.4.1 Practice

32.4.2 Extensions

32.4.3 Advanced

**33.1 Introduction**

33 Gamma Random Variables

33 Gamma Random Variables

33.2 Examples

33.3 Exercises

33.3.1 Practice

33.3.2 Extensions

33.3.3 Advanced

**34 Beta Random Variables**34.1 Introduction

34.2 Examples

34.3 Exercises

34.3.1 Practice

34.3.2 Extensions

**35 Normal Random Variables**

35.1 Introduction

35.2 Using the Normal Distribution: Scaling and Transforming to Standard Normal

35.3 \Backwards" Normal Problems

35.4 Summary: How to Distinguish a \Forward" Versus \Backwards" Normal Problem?

35.5 Exercises

35.5.1 Practice

35.5.2 Extensions

35.5.3 Advanced **36 Sums of Independent Normal Random Variables**36.1 The Sum of Independent Normal Random Variables is Normally Distributed

36.2 Why the Sum of Independent Normals is Normal Too (Optional)

36.3 Exercises

36.3.1 Practice

36.3.2 Extensions

36.3.3 Advanced

**37 Central Limit Theorem**

37.1 Introduction

37.2 Laws of Large Numbers

37.3 Central Limit Theorem

37.4 Applications of the Central Limit Theorem to Sums of Continuous Random Variables

37.5 Applications of the Central Limit Theorem to Sums of Discrete Random Variables

37.6 Normal Approximations to Binomial Random Variables

37.7 Normal Approximations to Poisson Random Variables

37.8 Exercises

37.8.1 Practice

37.8.2 Extensions

38 Review of Named Continuous Random Variables

38.1 Summing up: How do you tell all these random variables apart?

38.2 Exercises

**VII Additional Topics**

39 Variance of Sums; Covariance; Correlation

39.1 Introduction

39 Variance of Sums; Covariance; Correlation

39.2 Motivation for Covariance

39.3 Properties of the Covariance

39.4 Examples of Covariance

39.5 Linearity of the Covariance

39.6 Correlation

39.7 Exercises

39.7.1 Practice

39.7.2 Extensions

39.7.3 Advanced

**40.1 Introduction**

40 Conditional Expectation

40 Conditional Expectation

40.2 Examples

40.3 Exercises

40.3.1 Practice

40.3.2 Extensions

40.3.3 Advanced.

**41.1 Introduction**

41 Markov and Chebyshev Inequalities

41 Markov and Chebyshev Inequalities

41.2 Markov Inequality

41.3 Chebyshev Inequality

41.4 Exercises

41.4.1 Practice

41.4.2 Extensions

**42.1 Introduction**

42 Order Statistics

42 Order Statistics

42.2 Examples

42.3 Joint Density and Joint CDF of Order Statistics

42.4 Exercises

42.4.1 Practice

42.4.2 Extensions

43 Moment Generating Functions

43 Moment Generating Functions

43.1 A Brief Introduction to Generating Functions

43.2 Moment Generating Functions

43.3 Moment Generating Functions of Discrete Random Variables

43.4 Moment Generating Functions of Continuous Random Variables

43.5 Appendix: Building a Generating Function

43.6 Exercises

**44.1 Distribution of a Function of One Continuous Random Variable**

44 Transformations of One or Two Random Variables

44 Transformations of One or Two Random Variables

44.2 Joint Density of Two Random Variables That Are Functions of Another Pair of Random Variables

44.3 Exercises

44.3.1 Practice

44.3.2 Extensions

44.3.3 Advanced

45 Review Questions for All Chapters