Table Of ContentPraise for Who’s Counting?
uniting numbers and narratives With stories from
PoP Culture, Puzzles, PolitiCs, and more
“Who’s Counting? is an astonishing book, for it combines two things that
aren’t often found together: analytical rigor and fun. You don’t have to
be a math nerd to appreciate the fluid, easy explanations of everyday
innumerate reasoning, but I guarantee that this book will make you feel
smarter. Who’s Counting? is structured to enable the reader to dip in and
out, biting each chocolate to see what one likes, but I encourage you to
do what I did and just gorge yourself. It all tastes good.”—Lee McIntyre,
author of How to Talk to a Science Denier
“Math illiteracy breeds warped understandings of the world that ulti-
mately jeopardize the progress of civilization. Math professor John Allen
Paulos wrote Who’s Counting? as an entertaining, relevant, and potent
antidote to this societal blight.”—Neil deGrasse Tyson, astrophysicist,
American Museum of Natural History
“John Allen Paulos yet again defends logic and crusades against the prob-
lem of innumeracy with good humor and plain language. Using examples
ranging from math puzzles to controversial cultural issues, this engaging
book reminds us that citizens in a democracy must understand the facts
and figures that swirl about them every day.”—Tom Nichols, author of
The Death of Expertise
“What better than a mathematician sharing the intrigue of math with
us all and, at the same time, protecting us from the data manipulation
and fake news that is threatening our democracy. Who’s Counting? is a
must-read for all who want to wear mathematical armor against the
war on truth.”—Jo Boaler, Nominelli-Olivier Professor of Education
(Mathematics) at Stanford University, co-founder of youcubed.org,
and author of Limitless Mind
“I love Paulos’s writing. Within the ephemeral specifics of news, he finds
the enduring logic of mathematics. Who’s Counting? is a glimpse of a
more numerate world—and oh how I wish we lived there!”—Ben Orlin,
author of Math with Bad Drawings
“We too often pigeonhole ourselves as ‘number’ or ‘narrative’ people. But
Paulos breaks down that artificial division. With Who’s Counting? skill at
crafting numerate narratives is a superpower within our grasp.”—Philip
E. Tetlock, author of Superforecasting
“Who’s Counting? would not only make a super present for the math-
minded folks in your family (or for yourself), but also it provides a
rich resource for teachers to dive into for great ideas to spice up their
classes.”—Keith Devlin, emeritus mathematician and author, Stanford
University
Who’s Counting?
Uniting Numbers and Narratives with Stories from
Pop Culture, Puzzles, Politics, and More
John Allen Paulos
An imprint of Globe Pequot, the trade division of
The Rowman & Littlefield Publishing Group, Inc.
4501 Forbes Blvd., Ste. 200
Lanham, MD 20706
www.rowman.com
Distributed by NATIONAL BOOK NETWORK
Copyright © 2022 by John Allen Paulos
All rights reserved. No part of this book may be reproduced in any form or by any electronic or
mechanical means, including information storage and retrieval systems, without written permission
from the publisher, except by a reviewer who may quote passages in a review.
British Library Cataloguing in Publication Information Available
Library of Congress Cataloging-in-Publication Data
Names: Paulos, John Allen, author.
Title: Who’s counting? : uniting numbers and narratives with stories from pop culture, sports,
politics, and more / John Allen Paulos.
Description: Lanham, MD : Prometheus, an imprint of Globe Pequot, the trade division of The
Rowman & Littlefield Publishing Group, Inc., [2022] | Summary: “Who’s Counting features
selected columns from Paulos’s well-known ABC News series of the same name collected
here in book form for the first time, along with updates and brand-new original essays from
the author, to examine how better understanding data improves our thinking and decision-
making”—Provided by publisher.
Identifiers: LCCN 2022001235 (print) | LCCN 2022001236 (ebook) | ISBN 9781633888128
(paperback) | ISBN 9781633888135 (epub)
Subjects: LCSH: Mathematics—Popular works.
Classification: LCC QA93 .P374 2022 (print) | LCC QA93 (ebook) | DDC 510—dc23/
eng20220521
LC record available at https://lccn.loc.gov/2022001235
LC ebook record available at https://lccn.loc.gov/2022001236
The paper used in this publication meets the minimum requirements of American National
Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI/
NISO Z39.48-1992.
Dedicated to those who can be both
tough-minded and softhearted
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1: Puzzles as a Prelude . . . . . . . . . . . . . . . . . . . 1
The Monty Hall Puzzle: Variants of It and Its Connection
to a COVID-19 Precaution . . . . . . . . . . . . . . . . . . 3
Wanna Be President? Pass This Test. . . . . . . . . . . . . . . . 7
Now Featuring e: Pi Has Long Been in the Spotlight. What
about e?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Put on Your Hats and Codes: A Hat Puzzle and Error-
Correcting Codes. . . . . . . . . . . . . . . . . . . . . . . . 14
Scaling Up Is So Very Hard to Do . . . . . . . . . . . . . . . .17
Scaling in Biology, Metabolic Rates, and a Puzzle about
Evolutionary Time . . . . . . . . . . . . . . . . . . . . . . .19
Five or Six Reasons Why Parity Puzzles Are Fun. . . . . . . . . 22
Course Loads: Average Paradoxes That Went to College . . . . .25
Parrando’s Paradox: Losing plus Losing Equals Winning. . . . . 29
Chapter 2: A Bit of Probability . . . . . . . . . . . . . . . . . . .33
Probability and Independent Events: A Mini-Tutorial . . . . . .35
Collecting a Complete Set: Baseball Cards, Disease, and
Denial of Service . . . . . . . . . . . . . . . . . . . . . . . .39
Probability and Hitting Streaks: Does DiMaggio’s Deserve
an Asterisk?. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
How Many Ways: From Social Distancing to an Alternative
Solution to the Birthday Problem . . . . . . . . . . . . . . .44
Conditional Probability (Because We Usually Know Things)
and the Prosecutor’s Fallacy . . . . . . . . . . . . . . . . . .47
vii
Who’s Counting?
Four Times the Odds of Conviction Not Four Times as
Likely and Other Distinctions . . . . . . . . . . . . . . . . .48
Randonauts and Coincidences, Fun and Nonsense . . . . . . . .51
Calculating the Probability of Picking the Wrong Suspect . . . .54
COVID-19: False Positives, Fatality Rates, and Base Rates. . . . 57
Future World: Privacy, Terrorists, and Pre-Perpetrators . . . . . .62
Summer and Boys: A New, Very Counterintuitive Birthday
Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
Chapter 3: Lies and Logic . . . . . . . . . . . . . . . . . . . . . .67
Lying Brain Teasers: Politicians, Liars, and Mathematical
Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
Logical Liars, Paradoxical Politicians, and Smullyan’s
Stumper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Groucho Meets Russell . . . . . . . . . . . . . . . . . . . . . .74
True If and Only If False in the Market and in Life . . . . . . .77
Oh No. Denials and Conditional Statements Often
Counterproductive . . . . . . . . . . . . . . . . . . . . . . .79
The Ellsberg and St. Petersburg Paradoxes: Risks, Utility,
and Our Desire for Certainty. . . . . . . . . . . . . . . . . . 81
Justifiably Believing That Something Is True Doesn’t Mean
You Know It . . . . . . . . . . . . . . . . . . . . . . . . . .85
Prove This, Win $1,000,000! The Goldbach and Collatz
Conjectures and a Doable Puzzle. . . . . . . . . . . . . . . . 87
A Clever Card Trick and a Religious Hoax . . . . . . . . . . . .90
Does the Declaration of Independence Guarantee
Happiness? . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Weird Science from the Hume and Bacon Institute. . . . . . . . 94
Chapter 4: Calculations and Miscalculations . . . . . . . . . . . .97
From Dates and Y2K to People and PY2K: A Tale of Two
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
Apophenia and Spherical Cows: Clots and Coincidences. . . . 102
COVID, Calculus, and the Curve . . . . . . . . . . . . . . . . 104
Hoarding Medicines to Wearing Masks: Sociology, Public
Health, and the Prisoner’s Dilemma . . . . . . . . . . . . . 106
viii
Contents
Of Ants, Butterflies, and Economic Whimsy . . . . . . . . . . 109
There’s Nothing Wrong with Fuzzy Math. . . . . . . . . . . . 112
Predicting Success? SAT Scores and College Grades and
the Other SAT, the Soccer Assessment Test . . . . . . . . . 115
Be Careful What You Measure and Try to Achieve—You
Might Succeed . . . . . . . . . . . . . . . . . . . . . . . . 118
Sexonomics: Prostitutes’ Incomes, a Nonmoralistic Account . . 121
Sex, Lies, and Statistics: Some Musings . . . . . . . . . . . . . 123
Reductio: Abortion through the Looking Glass. . . . . . . . . 126
Misunderstandings: Jesus’ Genealogical Descendants,
Sexual Predators, and Home Run Records. . . . . . . . . . 129
Chapter 5: Partisanship in Politics. . . . . . . . . . . . . . . . . 133
Trump: Outrage Fatigue, Denial-of-Service Attacks, and
Brandolini’s Law of Refutation. . . . . . . . . . . . . . . . 134
Wolf’s Dilemma and Extreme Lockstep Political
Partisanship. . . . . . . . . . . . . . . . . . . . . . . . . . 136
Through a Mathnifying Glass Darkly . . . . . . . . . . . . . . 138
Ranked Choice Voting: More Welcoming to Moderate
and Less Conducive to Extreme Candidates. . . . . . . . . 140
Ties and Coin Flips: From Butterfly Ballots to Butterfly
Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Voting Blocs: Red States, Blue States, and a Model for
Thoughtless Voting . . . . . . . . . . . . . . . . . . . . . . 145
The Internet, Conspiracy Theories, and Cognitive Foibles,
Including the Conjunction Fallacy . . . . . . . . . . . . . . 147
After 9-11: Mindlessly Searching for Numerological
Meaning in the Midst of Tragedy . . . . . . . . . . . . . . 150
Chapter 6: Religious Dogmatism . . . . . . . . . . . . . . . . . 153
Naturally Evolving Order and Probability versus the Claims
of Intelligent Design . . . . . . . . . . . . . . . . . . . . . 154
On the Quasi Apotheosis of Mathematical Ideas, Two
Accounts—One Silly, the Other Serious. . . . . . . . . . . 158
Holy Cow or Bull? ELSes: It Would Be Astonishing If
One Didn’t Find Hidden Messages in the Bible. . . . . . . 161
ix
