Digital Dice: Computational Solutions to Practical Probability Problems

Digital Dice: Computational Solutions to Practical Probability Problems

by Paul J. Nahin

Paperback(With a New preface by the author)

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Overview

Some probability problems are so difficult that they stump the smartest mathematicians. But even the hardest of these problems can often be solved with a computer and a Monte Carlo simulation, in which a random-number generator simulates a physical process, such as a million rolls of a pair of dice. This is what Digital Dice is all about: how to get numerical answers to difficult probability problems without having to solve complicated mathematical equations.

Popular-math writer Paul Nahin challenges readers to solve twenty-one difficult but fun problems, from determining the odds of coin-flipping games to figuring out the behavior of elevators. Problems build from relatively easy (deciding whether a dishwasher who breaks most of the dishes at a restaurant during a given week is clumsy or just the victim of randomness) to the very difficult (tackling branching processes of the kind that had to be solved by Manhattan Project mathematician Stanislaw Ulam). In his characteristic style, Nahin brings the problems to life with interesting and odd historical anecdotes. Readers learn, for example, not just how to determine the optimal stopping point in any selection process but that astronomer Johannes Kepler selected his second wife by interviewing eleven women.

The book shows readers how to write elementary computer codes using any common programming language, and provides solutions and line-by-line walk-throughs of a MATLAB code for each problem.


Digital Dice will appeal to anyone who enjoys popular math or computer science. In a new preface, Nahin wittily addresses some of the responses he received to the first edition.

Product Details

ISBN-13: 9780691158211
Publisher: Princeton University Press
Publication date: 03/24/2013
Series: Princeton Puzzlers
Edition description: With a New preface by the author
Pages: 288
Sales rank: 1,240,734
Product dimensions: 5.50(w) x 8.40(h) x 0.90(d)

About the Author

Paul J. Nahin is the author of many best-selling popular-math books, including Chases and Escapes, Dr. Euler's Fabulous Formula, When Least is Best, Duelling Idiots and Other Probability Puzzlers, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.

Table of Contents

Preface to the Paperback Edition xiii

Introduction 1




The Problems 35

1. The Clumsy Dishwasher Problem 37

2. Will Lil and Bill Meet at the Malt Shop? 38

3. A Parallel Parking Question 40

4. A Curious Coin-Flipping Game 42

5. The Gamow-Stern Elevator Puzzle 45

6. Steve's Elevator Problem 48

7. The Pipe Smoker's Discovery 51

8. A Toilet Paper Dilemma 53

9. The Forgetful Burglar Problem 59

10. The Umbrella Quandary 61

11. The Case of the Missing Senators 63

12. How Many Runners in a Marathon? 65

13. A Police Patrol Problem 69

14. Parrondo's Paradox 74

15. How Long Is the Wait to Get the Potato Salad? 77

16. The Appeals Court Paradox 81

17. Waiting for Buses 83

18. Waiting for Stoplights 85

19. Electing Emperors and Popes 87

20. An Optimal Stopping Problem 91

21. Chain Reactions, Branching Processes, and Baby Boys 96



MATLAB Solutions To The Problems 101

1. The Clumsy Dishwasher Problem 103

2. Will Lil and Bill Meet at the Malt Shop? 105

3. A Parallel Parking Question 109

4. A Curious Coin-Flipping Game 114

5. The Gamow-Stern Elevator Puzzle 120

6. Steve's Elevator Problem 124

7. The Pipe Smoker's Discovery 129

8. A Toilet Paper Dilemma 140

9. The Forgetful Burglar Problem 144

10. The Umbrella Quandary 148

11. The Case of the Missing Senators 153

12. How Many Runners in a Marathon? 157

13. A Police Patrol Problem 160

14. Parrondo's Paradox 169

15. How Long is the Wait to Get the Potato Salad? 176

16. The Appeals Court Paradox 184

17. Waiting for Buses 187

18. Waiting for Stoplights 191

19. Electing Emperors and Popes 197

20. An Optimal Stopping Problem 204

21. Chain Reactions, Branching Processes, and Baby Boys 213



Appendix 1. One Way to Guess on a Test 221

Appendix 2. An Example of Variance-Reduction in the Monte Carlo Method 223

Appendix 3. Random Harmonic Sums 229

Appendix 4. Solving Montmort's Problem by Recursion 231

Appendix 5. An Illustration of the Inclusion-Exclusion Principle 237

Appendix 6. Solutions to the Spin Game 244

Appendix 7. How to Simulate Kelvin's Fair Coin with a Biased Coin 248

Appendix 8. How to Simulate an Exponential Random Variable 252

Appendix 9. Index to Author-Created MATLAB m-Files in the Book 255



Glossary 257

Acknowledgments 259

Index 261

Also by Paul J. Nahin 265

What People are Saying About This

Nick Hobson

One of the strengths of Digital Dice is its wealth of historical information. Nahin carefully notes the origin of each problem and traces its history. He also tells a number of amusing anecdotes. I found all the problems interesting, especially Parrondo's Paradox. Anyone who has not met this paradox will be amazed by it! Digital Dice is a very enjoyable read.
Nick Hobson, creator of the award-winning Web site Nick's Mathematical Puzzles

Martin Gardner

Paul Nahin's Digital Dice is a marvelous book, one that is even better than his Duelling Idiots. Nahin presents twenty-one great probability problems, from George Gamow's famous elevator paradox (as corrected by Donald Knuth) to a bewildering puzzle involving two rolls of toilet paper, and he solves them all with the aid of Monte Carlo simulations and brilliant, impeccable reasoning.

Julian Havil

By presenting problems for which complete theoretical analysis is difficult or currently impossible, Digital Dice is a reminder that mathematics is often advanced by investigation, long before theoretical tools are brought to bear. The book's choice of problems is eclectic and interesting, and the explanations are clear and easy to read. A welcome addition to popular mathematical literature.
Julian Havil, author of "Nonplussed!: Mathematical Proof of Implausible Ideas"

Joseph Mazur

Open this delightful, matchless book to be sucked into a treasure trove of wonderful conundrums of everyday life. Then, persuaded by straightforward Monte Carlo simulation exercises, emerge refreshed, invigorated, and fully satisfied by the unique experience of learning from Nahin's marvelous Digital Dice.
Joseph Mazur, author of "The Motion Paradox"

From the Publisher

"Nahin's new book is a rich source of tantalizing, real-life probability puzzles that require considerable ingenuity, and in most cases computer simulation, to solve. Though written to be delved into rather than read cover-to-cover, Digital Dice has an engaging and often witty style that makes each chapter a pleasurable read."—Keith Devlin, author of The Math Gene and The Math Instinct

"Paul Nahin's Digital Dice is a marvelous book, one that is even better than his Duelling Idiots. Nahin presents twenty-one great probability problems, from George Gamow's famous elevator paradox (as corrected by Donald Knuth) to a bewildering puzzle involving two rolls of toilet paper, and he solves them all with the aid of Monte Carlo simulations and brilliant, impeccable reasoning."—Martin Gardner

Keith Devlin

Nahin's new book is a rich source of tantalizing, real-life probability puzzles that require considerable ingenuity, and in most cases computer simulation, to solve. Though written to be delved into rather than read cover-to-cover, Digital Dice has an engaging and often witty style that makes each chapter a pleasurable read.
Keith Devlin, author of The "Math Gene" and "The Math Instinct"

Customer Reviews