Benedict Gross - Fat Chance: Probability from 0 to 1

In a world where we are constantly being asked to make decisions based on incomplete information, facility with basic probability is an essential skill. This book provides a solid foundation in basic probability theory designed for intellectually curious readers and those new to the subject. Through its conversational tone and careful pacing of mathematical development, the book balances a charming style with informative discussion.

This text will immerse the reader in a mathematical view of the world, giving them a glimpse into what attracts mathematicians to the subject in the first place. Rather than simply writing out and memorizing formulas, the reader will come out with an understanding of what those formulas mean, and how and when to use them. Readers will also encounter settings where probabilistic reasoning does not apply or where our intuition can be misleading. This book establishes simple principles of counting collections and sequences of alternatives and elaborates on these techniques to solve real-world problems both inside and outside the casino. Readers at any level are equipped to consider probability at large and work through exercises on their own.

BENEDICT GROSS is Leverett Professor of Mathematics, Emeritus at Harvard University and Professor of Mathematics at UC San Diego. He has taught mathematics at all levels at Princeton, Brown, Harvard, and UCSD, and served as the Dean of Harvard College from 20032007. He is a member of the American Academy of Arts and Sciences and the National Academy of Science. Among his awards and honors are the Cole Prize from the American Mathematical Society and a MacArthur Fellowship. His research is primarily in number theory.

JOE HARRIS is the Higgins Professor of Mathematics at Harvard University. He has been at Harvard since 1988 and was previously on the faculty at MIT and Brown. He is a member of the American Academy of Arts and Sciences and the National Academy of Science. Throughout his career, he has been deeply committed to education at every level, which led to a partnership with Benedict Gross to develop the Harvard course Fat Chance, the inspiration for the book of the same title. He is author of several books including 3264 and All That, Algebraic Geometry, and The Geometry of Schemes.

EMILY RIEHL is an Assistant Professor of Mathematics at Johns Hopkins University and previously was a Benjamin Peirce and NSF postdoctoral fellow at Harvard University. She has published over twenty papers and written two books: Categorical Homotopy Theory and Category Theory in Context. She has been awarded an NSF grant and a CAREER award to support her work and has been recognized for excellence in teaching at both Johns Hopkins and Harvard.

BENEDICT GROSS

Harvard University

JOE HARRIS

Harvard University

EMILY RIEHL

Johns Hopkins University

University Printing House, Cambridge CB2 8BS, United Kingdom

One Liberty Plaza, 20th Floor, New York, NY 10006, USA

477 Williamstown Road, Port Melbourne, VIC 3207, Australia

314321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi 110025, India

79 Anson Road, #0604/06, Singapore 079906

Cambridge University Press is part of the University of Cambridge.

It furthers the Universitys mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence.

www.cambridge.org

Information on this title: www.cambridge.org/9781108482967

DOI: 10.1017/9781108610278

Benedict Gross, Joe Harris, and Emily Riehl 2019

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2019

Printed in the United Kingdom by TJ International Ltd., Padstow, Cornwall

A catalogue record for this publication is available from the British Library.

Library of Congress Cataloging-in-Publication Data

Names: Gross, Benedict H., 1950- author. | Harris, Joe, 1951 author. | Riehl, Emily, author.

Title: Fat chance : probability from 0 to 1 / Benedict Gross (Harvard University, Massachusetts), Joe Harris (Harvard University, Massachusetts), Emily Riehl (Johns Hopkins University).

Description: Cambridge ; New York, NY : Cambridge University Press, c2019. |

Includes bibliographical references and index.

Identifiers: LCCN 2018058461| ISBN 9781108482967 (hardback : alk. paper) |

ISBN 9781108728188 (pbk. : alk. paper)

Subjects: LCSH: ProbabilitiesPopular works. | ProbabilitiesProblems, exercises, etc.

Classification: LCC QA273.15 .G76 2019 | DDC 519.2dc23

LC record available at https://lccn.loc.gov/2018058461

ISBN 978-1-108-48296-7 Hardback

ISBN 978-1-108-72818-8 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Suppose a friend comes up to you in a bar with the following challenge. He asks you for a quarterwhich you hand over somewhat reluctantlyand tells you hes going to flip it six times and record the outcome: heads or tails. Your friend then offers you the following bet: if you can correctly predict the number of heads, hell buy the next round of drinks. But if the outcome is different from what you guessed, the next round is on you. On each individual flip heads and tails are equally likely. So it seems that three heads and three tails is the most likely outcome. But is this a good bet?

Or, you and a friend are driving to the movies and find a place to park a mile away from the theater. Its a busy area, but you know theres some chance that if you drive on, youll find a space thats only half a mile away. Its also possible that if you look for a better spot, youll have to come back to this one, which might no longer be there. How do you decide what to do? And how does it affect your decision if the movie starts in 25 minutes?

In a school election, a vast majority of the left-handed voters prefer Tracy, who has advocated strongly for more left-handed desks to be put in classrooms. Does this mean that Tracy is likely to win? And while were at it, what does a 3% margin of error in a poll really mean?

Or suppose you go to the doctor with your uncle and are told that his level of prostate-specific antigen (PSA) is high. The doctor reports that a large proportion of people with prostate cancer have elevated PSA levels. How worried should you be?

Finally, suppose you want to gamble with your lifes savings, totaling $1,000, while playing roulette. Is it better to wager it all on a single bet, or to bet just $1 at a time until you either double your money or go bankrupt?

At its heart, probability is about decision-making, at least in contexts when its possible to determine how mathematically likely a given event would be. In this book we embark upon a journey that will allow us to discover the answers to all of the questions posed above, using the mathematics of probability theory to help us identify the ways in which your intuition may lead you astray.