Will You Be Alive 10 Years from Now?: And Numerous Other Curious Questions in Probability

262 Pages·2013·3.77 MB·English

by  Paul J. Nahin

Will You Be Alive
10 Years from Now?
Will You Be Alive
10 Years from Now?
And Numerous Other Curious
Questions in Probability
A collection of not so well-known mathematical mind-benders
(with solutions, with one exception)
PAUL J. NAHIN
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2014 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
press.princeton.edu
Jacket art by Dimitri Karetnikov All Rights Reserved
Library of Congress Cataloging-in-Publication Data Nahin, Paul J.
Will you be alive 10 years from now? : and numerous other curious questions in
probability : a collection of not so well-known mathematical mind-benders (with
solutions, with one exception) / Paul J. Nahin.
pages cm Summary: “What are the chances of a game-show contestant finding a chicken
in a box? Is the Hanukkah dreidel a fair game? Will you be alive ten years from now?
These are just some of the one-of-a-kind probability puzzles that acclaimed popular
math writer Paul Nahin offers in this lively and informative book. Nahin brings
probability to life with colorful and amusing historical anecdotes as well as an
electrifying approach to solving puzzles that illustrates many of the techniques that
mathematicians and scientists use to grapple with probability. He looks at classic
puzzles from the past—from Galileo’s dice-tossing problem to a disarming dice puzzle
that would have astonished even Newton—and also includes a dozen challenge
problems for you to tackle yourself, with complete solutions provided in the back of the
book. Nahin then presents twenty-five unusual probability puzzlers that you aren’t
likely to find anywhere else, and which range in difficulty from ones that are easy but
clever to others that are technically intricate. Each problem is accompanied by an
entertaining discussion of its background and solution, and is backed up by theory and
computer simulations whenever possible in order to show how theory and computer
experimentation can often work together on probability questions. All the MATLAB
Monte Carlo simulation codes needed to solve the problems computationally are
included in the book. With his characteristic wit, audacity, and insight, Nahin
demonstrates why seemingly simple probability problems can stump even the
experts.”—Provided by publisher.
Includes bibliographical references and index.
ISBN 978-0-691-15680-4 (hardback) 1. Probabilities—Problems, exercises, etc. I. Title.
QA273.25.N344 2014
519.2—dc23 2013010221
British Library Cataloging-in-Publication Data is available This book has been composed
in ITC New Baskerville Printed on acid-free paper. ∞
Printed in the United States of America 1 3 5 7 9 10 8 6 4 2
For Patricia Ann
who, after fifty years of marriage, can still surprise me
even more than a good probability problem can!
“If we do everything right, if we do it with
absolute certainty, there’s still a 30 percent
chance we’re going to get it wrong.”
— Joseph R. Biden, vice president of the United
States, during a 2009 political talk in
Williamsburg, Virginia, that showed a shaky
understanding of probability is no roadblock to
higher elective office in America
What has hit Los Angeles is a force of nature that is exactly the same as a
flood or a fire or an earthquake. It’s the natural law of probability. Given
four to five million vehicles traveling on those freeways each day, a
certain number of accidents will happen. It was mathematically unlikely
that one would happen at each of seventeen major interchanges
simultaneously on any given day, but the odds caught up with us.
Mathematics is a force of nature, too.
From Thomas Perry’s very funny 1983 novel
Metzger’s Dog, in which LA thieves show that the
CIA’s middle initial might be claiming just a bit
too much
There are few persons, even among the calmest thinkers, who have not
occasionally been startled into a vague yet thrilling half-credence in the
supernatural, by coincidences of so seemingly marvelous a character that,
as mere coincidences, the intellect has been unable to receive them. …
Such sentiments are seldom thoroughly stifled unless by reference to the
doctrine of chance, or as it is technically termed, the Calculus of
Probabilities. Now this Calculus is, in its essence, purely mathematical;
and thus we have the anomaly of the most rigidly exact in science
applied to the shadow and spirituality of the most intangible in
speculation.
The opening words of Edgar Allen Poe’s
“The Mystery of Marie Rogêt” (1842/3)
Contents

Preface xv

Introduction: Classic Puzzles from the Past
1
I.1 A Gambling Puzzle of Gombaud and Pascal 1
I.2 Galileo’s Dice Problem 3
I.3 Another Gombaud-Pascal Puzzle 4
I.4 Gambler’s Ruin and De Moivre 6
I.5 Monte Carlo Simulation of Gambler’s Ruin 10
I.6 Newton’s Probability Problem 13
I.7 A Dice Problem That Would Have Surprised Newton 17
I.8 A Coin-Flipping Problem 18
I.9 Simpson’s Paradox, Radio-Direction Finding, and the
Spaghetti Problem 21

Challenge Problems
30
1
Breaking Sticks
36
1.1 The Problem 36
1.2 Theoretical Analysis 36
1.3 Computer Simulation 38