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b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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World Scientific
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Control Number: 2021003538
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
CHANCE, LOGIC AND INTUITION
An Introduction to the Counter-Intuitive Logic of Chance
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
ISBN 978-981-122-918-3 (hardcover)
ISBN 978-981-122-919-0 (ebook for institutions)
ISBN 978-981-122-920-6 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12069#t=suppl
Desk Editor: Soh Jing Wen
Printed in Singapore
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“And thus, by combining the uncertainty of chance with the force of
mathematical proof and by the reconciliation of two apparent opposites,
she derives her name from both of them and rightfully assumes the
wonderful name of Mathematics of chance!” — Blaise Pascal
v
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Preface
Few people have won the Nobel Prize twice. Linus Pauling (1901–94) is
one of them. The American chemist and peace activist was awarded the
Nobel Prize in Chemistry in 1954 and followed this up with the Nobel
Peace Prize seven years later. During the ceremony he told the audience
that winning the Nobel Prize for the second time is not as exceptional as
it seems. It’s true, he admitted, that the chances of anyone winning a
Nobel Prize for the first time are very small. But the chances of winning
a second Nobel Prize are much better. After all, you only have to
compete with others who have already won a Nobel Prize and who are
still alive.
Pauling was joking, of course. But the anecdote illustrates how easily
we are misled when it comes to probabilities. As Persi Diaconis, a well-
known expert on probability, once put it: “Our brains are just not wired
to do probability problems very well.” When asked about the surprising
solution to a famous probability problem known as the Monty Hall
dilemma, he admitted that his intuitive response to similar problems had
been wrong time and time again. When it comes to probabilities, our
intuition is a poor guide.
Our intuition solves problems by taking shortcuts. Instead of analyzing
the problem, it automatically translates it, so to speak, into a similar-
looking problem with a familiar solution. It then solves the original
problem by analogy. In psychology this intuitive form of thinking is
qualified as system 1 thinking. Daniel Kahneman, another Nobel Prize
winner and known for his work on the psychology of judgement, refers to
it as fast thinking.
vii
viii Chance, Logic and Intuition
Fast thinking often works well enough, but not in the realm of
chance, where familiar principles typically do not apply. Take, for
example, the rolling of a die. In contrast to what many people intuitively
expect, the probability of rolling a six doesn’t increase with every
unsuccessful roll of the die. Or, say, you bet on rolling at least one six in
two consecutive rolls. In this case the overall chance is not the sum of the
individual probabilities; it’s less. When you’re dealing with possibilities,
one and one don’t add up to two.
These misconceptions may betray a naïve understanding of probability,
but also on a more advanced level of understanding our intuition tends to
steer us in the wrong direction. In fact, we often find ourselves at a
complete loss in the face of a probability problem. And yet the realm of
chance is not entirely inaccessible, as the ancient Greek philosopher
Aristotle once thought. The unpredictability of chance has a logic of its
own, a logic that is governed by the rules of probability. So, can
probability theory help us where our intuition fails? It certainly can, but in
this regard a different kind of obstacle emerges.
Of all the branches of mathematics, probability theory is perhaps the
closest to our daily lives. But if you open a standard book on probability,
you may very well feel far removed from the reality of daily life.
Mathematical symbols and formulas stare back at you with almost every
turn of the page. Little is mentioned about the role of chance in our
everyday lives. And even if you’re not scared off by some abstract math,
you may still be disappointed and close the book, never to open a book
on probability again.
This book takes an entirely different approach. Instead of a formal
presentation of probability, packed with formulas and definitions, the
mathematics has been embedded into an everyday narrative context.
Part One tells the story of the “taming of chance”. You will meet the
pioneers of the mathematics of probability. Contrary to popular belief,
these mathematicians did not live their lives secluded in a Platonic world
of mathematical ideas. They kept a firm eye on the world around them.
They also had their own dreams. Understanding how chance operates can
free us from the notion that we are at the whim of a fickle goddess, or
Preface ix
that we are fettered by the chains of fate. It can make us fair gamblers. It
can even guide us through life’s uncertainties.
These pioneering mathematicians were fascinating personalities, in no
way inferior to their famous contemporaries in literature and philosophy.
Some are still known, such as Galileo Galilei and Blaise Pascal, while
others are almost forgotten, as, for instance, the colorful scholar Girolamo
Cardano. Hopefully, this story about the birth of probability will bring
them back to life.
What about the math in the book? You don’t have to be a diehard
mathematician to understand the mathematical parts of this founding story.
The elementary skills of adding, subtracting, dividing, and multiplying
numbers will do. By following in the footsteps of the pioneers of
probability, you will learn, in a natural way, some basic concepts and
simple rules of probability. At the same time, you will also learn to avoid
the pitfalls that the founders of probability encountered in their path.
In Part Two we will explore the counter-intuitive logic of chance. Chance
impacts everyday life in many different ways. The future is often uncertain
and unpredictable, and our path is full of coincidences. You would expect
us to have some understanding of the capricious ways of chance, but that is
far from being the case. When we face chance in everyday life our
intuition fools us time and again.
Some misconceptions have even been given their own names, such as
the fallacy of the transposed conditional or, the most deceitful of all, the
gambler’s fallacy. They are called fallacies because their deceptive
reasoning appears to be logical. It isn’t. Despite seeming very convincing,
they continually steer us in the wrong direction.
When we begin to truly understand chance it often appears paradoxical
and hard to believe — not because it is really paradoxical, but because, at
first glance, it runs completely counter to our intuition. However, it will
stop looking like this once we have replaced our intuitive framing with an
alternative way of looking.
Unfortunately, many of our intuitive misconceptions of the way chance
operates are deeply rooted. They can only be eradicated by thinking slow.
But equipped with a basic understanding of the rules of probability and a
good dose of inventiveness, you will be able to rid yourself of these
