Digital Dice: Computational Solutions to Practical Probability Problems

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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.

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Editorial Reviews

American Scientist - Brian Hayes
The problems are accessible but still realistic enough to be engaging, and the solutions in the back of the book will get you through any sticky spots. Writing your own versions of a few of these programs will acquaint you with a useful approach to problem solving and a novel style of thinking.
Mathematics Today - Alan Stevens
[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics.
Mathematical Reviews - G A. Hewer
After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences.
Mathematics Teacher - Thomas Sonnabend
This well-written entertaining collection of twenty-one probability problems presents their origin and history as well as their computer solutions. . . . These problems could be used in a computer programming course or a probability course that includes Monte Carlo simulations.
The UMAP Journal - James M. Cargal
All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teaching undergraduates in mathematics the fundamentals of computation and simulation.
From the Publisher

"The problems are accessible but still realistic enough to be engaging, and the solutions in the back of the book will get you through any sticky spots. Writing your own versions of a few of these programs will acquaint you with a useful approach to problem solving and a novel style of thinking."--Brian Hayes, American Scientist

"[T]he book is targeted at teachers and students of probability theory or computer science, as well as aficionados of recreational mathematics, but anyone who is familiar with the basics of probability and is capable of writing simple computer programs will have no problem working their way through this interesting and rewarding book."--Physics World

"[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics."--Alan Stevens, Mathematics Today

"Digital Dice will appeal to recreational mathematicians who have even a limited knowledge of computer programming, and even nonprogrammers will find most of the problems entertaining to ponder."--Games Magazine

"After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences."--G A. Hewer, Mathematical Reviews

"This well-written entertaining collection of twenty-one probability problems presents their origin and history as well as their computer solutions. . . . These problems could be used in a computer programming course or a probability course that includes Monte Carlo simulations."--Thomas Sonnabend, Mathematics Teacher

"All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teaching undergraduates in mathematics the fundamentals of computation and simulation."--James M. Cargal, The UMAP Journal

American Scientist
The problems are accessible but still realistic enough to be engaging, and the solutions in the back of the book will get you through any sticky spots. Writing your own versions of a few of these programs will acquaint you with a useful approach to problem solving and a novel style of thinking.
— Brian Hayes
Physics World
[T]he book is targeted at teachers and students of probability theory or computer science, as well as aficionados of recreational mathematics, but anyone who is familiar with the basics of probability and is capable of writing simple computer programs will have no problem working their way through this interesting and rewarding book.
Mathematics Today
[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics.
— Alan Stevens
Games Magazine
Digital Dice will appeal to recreational mathematicians who have even a limited knowledge of computer programming, and even nonprogrammers will find most of the problems entertaining to ponder.
Mathematical Reviews
After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences.
— G A. Hewer
Mathematics Teacher
This well-written entertaining collection of twenty-one probability problems presents their origin and history as well as their computer solutions. . . . These problems could be used in a computer programming course or a probability course that includes Monte Carlo simulations.
— Thomas Sonnabend
The UMAP Journal
All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teaching undergraduates in mathematics the fundamentals of computation and simulation.
— James M. Cargal
Games Magazine
Digital Dice will appeal to recreational mathematicians who have even a limited knowledge of computer programming, and even nonprogrammers will find most of the problems entertaining to ponder.
Mathematical Reviews
After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences.
— G A. Hewer
Mathematics Today
[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics.
— Alan Stevens
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Product Details

  • ISBN-13: 9780691126982
  • Publisher: Princeton University Press
  • Publication date: 3/3/2008
  • Pages: 276
  • Product dimensions: 6.41 (w) x 9.40 (h) x 0.94 (d)

Meet 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.

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Table of Contents

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 264

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