Taking Pascal's Wager By Michael Rota

8182019 Taking Pascals Wager By Michael Rota

httpslidepdfcomreaderfulltaking-pascals-wager-by-michael-rota 124

M I C H A E L R OTA

T A K I N G

P A S C A L rsquo S

W A G E R

F A I T H E V I D E N C E and the A B U N D A N T L I F E





T A K I N G

P A S C A L rsquo S

W A G E R


MI CH AE L ROTA

Copyrighted Material wwwivpresscompermissions



InterVarsity Press

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copy983090983088983089983094 by Michael Rota

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information about local and regional activities visit intervarsityorg

Scripture quotations unless otherwise noted are from the New Revised Standard Version of the Bible copyright

983089983097983096983097 by the Division of Christian Education of the National Council of the Churches of Christ in the USA Used by

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ISBN 983097983095983096-983088-983096983091983088983096-983093983089983091983094-983090 (print)

ISBN 983097983095983096-983088-983096983091983088983096-983097983097983097983097-983097 (digital)

Printed in the United States of America

As a member of the Green Press Initiative InterVarsity Press is committed to protecting

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Library of Congress Cataloging-in-Publication Data

Names Rota Michael 983089983097983095983093- author

itle aking Pascalrsquos wager faith evidence and the abundant life

Michael Rota

Description Downers Grove InterVarsity Press 983090983088983089983094 | Includes

bibliographical references and index

Identifiers LCCN 983090983088983089983094983088983088983095983097983090983090 (print) | LCCN 983090983088983089983094983088983089983088983094983094983090 (ebook) | ISBN

983097983095983096983088983096983091983088983096983093983089983091983094983090 (pbk alk paper) | ISBN 983097983095983096983088983096983091983088983096983097983097983097983097983097 (eBook)

Subjects LCSH Faith and reason--Christianity | Apologetics | Pascal

Blaise 983089983094983090983091-983089983094983094983090 Pensacircees | Philosophical theology

Classification LCC B983093983088 R983094983092983093 983090983088983089983094 (print) | LCC B983093983088 (ebook) | DDC 983090983091983097--dc983090983091

LC record available at httplccnlocgov983090983088983089983094983088983088983095983097983090983090

P 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094 983089983093 983089983092 983089983091 983089983090 983089983089 983089983088 983097 983096 983095 983094 983093 983092 983091 983090 983089

Y 983091983093 983091983092 983091983091 983091983090 983091983089 983091983088 983090983097 983090983096 983090983095 983090983094 983090983093 983090983092 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094




CONTENTS

Acknowledgments 983097

Introduction 10486251048625

P983137983154983156 983089 U983150983139983141983154983156983137983145983150983156983161 983137983150983140 C983151983149983149983145983156983149983141983150983156

1048625 A Curious Oer 9830901048625

983090 Pascalrsquos Wager he Basic Argument 9830911048624

983091 Objections to the Wager Moral Reservations and

the Cost o Commitment 983093983090

983092 More Objections to the Wager Other Religions

and Christianity 983094983093

P983137983154983156 983090 E983158983145983140983141983150983139983141

983093 Where Did Physical hings Come From 983096983091

983094 Why Is the Universe Just Right or Lie 983097983097

983095 A Primer on Probability 10486251048625983091

983096 God and the Multiverse 1048625983090983093

983097 he Beauty and Existential Resonance o Christianity 1048625983091983093

10486251048624 Counterevidence Divine Hiddenness and Evil 1048625983092983091

10486251048625 Historical Evidence or Christianity he Resurrection 1048625983093983092

1048625983090 Miracle or Myth 1048625983094983093

P983137983154983156 983091 S983137983161983145983150983143 Y983141983155 983156983151 G983151983140

1048625983091 Dietrich Bonhoeer 104862598309610486251048625983092 Jean Vanier 1048625983097983093

1048625983093 Immaculeacutee Ilibagiza 98309010486251048625

Conclusion aking the Wager 9830909830901048625




Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095

Scripture Index 983090983092983097




INTRODUCTION

I983155 983145983156 983154983137983156983145983151983150983137983148 983156983151 983138983141983148983145983141983158983141 983145983150 G983151983140983103 Is it reasonable to commit

onesel to Godmdashto live a deeply religious lie And how much certainty does

one need beore the time has come to decide I yoursquove pondered these ques-

tions beore this book is or you

I was raised Catholic although religion was not a major part o our amily

lie We prayed beore meals and went to church on most Sundays but there

was little discussion o religious matters and no regular personal prayer atleast not on my part Sometime in my early teens I began to wonder about

the truth o the religion I had been born into Does God really exist I he

does then relationship with God is the most important part o human lie

But i he doesnrsquot then the religious person is enmeshed in a massive de-

ception ldquoWe believe in one God the Father the Almighty rdquo I ound

mysel saying at Mass But did I really mean it I wasnrsquot sure and the ear

that I was being dishonest concerned meI continued to wonder about God and the rationality o Christianity

throughout high school in college and in graduate school Now as a pro-

essor o philosophy itrsquos my job to think about these topics every day My

research ocuses on the rationality o religious belie and I wrestle with ar-

guments or and against God with classes ull o students rom every point

o view almost every semester I have become convinced that it is rational

to live a deeply religious Christian lie Indeed it may be irrational not to

When contemplating the choice to commit to living a Christian lie one

might suppose that one should rerain rom making a commitment in the

absence o rock-solid evidence or the truth o Christianity Reflection on

personal relationships suggests otherwise In Te Will to Believe William




852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

James asks us to consider a man who hesitates ldquoindefinitely to ask a certain

woman to marry him because he [is] not perectly sure that she would

prove [to be] an angel afer he brought her home Would he not cut himseloff rom that particular angel-possibility as decisively as i he went and

married some one elserdquo1 In the area o romantic love it can be reasonable

to invest deeply in a relationship and eventually make a lielong com-

mitment to the beloved despite the absence o airtight evidence that the

marriage will be a happy one Because there is so much at stake it can be

reasonable to make a commitment to a personal relationship even when

absolute certainty proves elusive Applying this to the question o Godeven i the evidence or God lef some room or doubt considerations

about the possible value o a relationship with God might avor the decision

to make a religious commitment

Seventeenth-century French mathematician and theologian Blaise Pascal

gave an argument that expands on this insight Pascalrsquos wager as the ar-

gument has come to be known addresses those who arenrsquot sure whether

Christianity is true but think that it might be true Te argument can besummed up in a single sentence It is rational to seek a relationship with God

and live a deeply Christian lie because there is very much to gain and rela-

tively little to lose

As the wager is usually presented whatrsquos to gain is eternal happiness or

the wagerer An alternative and more powerul version o the argument

however ocuses not just on sel-interest but also on goods that go beyond

sel-interest I one commits to God and God does in act exist one brings

joy to God and one is better able to help others attain union with God And

i Jesus really is who he claimed to be we may even have a moral duty to

commit to living a Christian lie I Christianity is true we have a duty to

love God and have been called by God himsel who has given us everything

good that we have to live a deeply religious lie More than just sel-interest

can motivate one to take the wager

On the other hand i Christianity is alse the committed Christian has

still lived a meaningul lie has pursued moral excellence and has enjoyed

the many empirically well-attested benefits o belonging to a religious com-

munity Much to gain relatively little to lose

In part one o this book (chapters one through our) Irsquoll present an updated





version o Pascalrsquos wager strengthened by cutting-edge research rom psy-

chologists sociologists and philosophers Afer introducing and laying out

the basic argument in chapters one and two I turn to objections to the wagerin chapters three and our Tese include the objections that committing to

God on the basis o pragmatic considerations is immoral that the cost o

religious commitment is too high that the existence o religions other than

Christianity nullifies the argument and that Christian doctrine itsel casts

doubt on the wager When addressing this last issue I discuss grace ree will

and predestination

In part two (chapters five through twelve) Irsquoll take a careul look at argu-ments or the existence o God and the truth o Christianity showing howmdash

once we no longer demand certaintymdashthe available evidence is sufficient to

make serious Christian commitment entirely reasonable ogether parts

one and two present the bookrsquos main argument which can be summarized

as ollows

I Christianity has at least a 1048629983088 percent chance o being true then it is rational

to commit to living a Christian lie (the conclusion o part one)

Christianity does have at least a 1048629983088 percent chance o being true (the con-

clusion o part two)

Tus it is rational to commit to living a Christian lie

Te argument o part two proceeds in two stages First I present evidence

or theism (the view that there is a God) second I present evidence or the

more specific view o Christian theism I begin by asking the question o whyphysical reality exists at all Chapter five argues that there is at least one nec-

essarily existing being which explains the existence o contingent beings

(Contingent beings are things that reality didnrsquot have to include like all the

physical objects we see around us) Chapters six through eight then provide

an argument that the cause o physical reality is an intelligent being In the

last several decades mainstream physicists and astronomers have come to

realize that the lie-permitting character o our universe is balanced on a

kniersquos edge i several eatures had been ever-so-slightly different rom what

they in act are then stable sel-reproducing lie would not have been able

to arise Chapter six introduces some o the scientific evidence or this con-

clusion and gives a preliminary statement o what has come to be known as




852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

the fine-tuning argument Tis argument has been summarized or non-

specialists in several places but not always with enough rigor and back-

ground or the strength o the case to be ully displayed In chapter seven Iexplain the parts o probability theory required to appreciate the power o

the evidence that our universe is the product o an intelligence Chapter eight

contains an original reply to the strongest alternative to design the multi-

verse hypothesis (Tis is the hypothesis that our universe is just one o a vast

number o universes most o which are not lie permitting) aken together

chapters five through eight provide evidence or God drawing on the best

recent research but written so as to be accessible to the general readerIn chapter nine I turn to specifically Christian doctrines suggesting that

the beauty and existential resonance o Christianity are clues to its truth

Chapter ten concerns the two most powerul arguments against theism the

argument rom divine hiddenness and the argument rom evil (As argu-

ments against all orms o theism they are arguments against Christian

theism as well) Tese topics deserve not a single chapter but whole books

o their own And they have them So in chapter ten I summarize what I taketo be the strongest replies to the arguments rom hiddenness and evil o-

cusing on the recent work o Peter van Inwagen and Eleonore Stump Finally

in chapters eleven and twelve I turn to arguments or the resurrection o

Jesus Tis is just one o many Christian doctrines o course but logically

speaking it holds a special place I one has reason to believe that Jesus rose

rom the dead then one has reason to believe that Jesusrsquo teachings have

the divine stamp o approval and are thereore true So an argument or the

resurrection is an argument or the truth o Christianity (or at least the

larger part o one) In this pair o chapters I draw on the most recent work

on the issue by historians theologians and philosophers While I think that

the evidence o part two is by itsel sufficient to justiy belie in Christianity

agreement on this point is not required or the main argument o the book

to succeed For given the argument o part one all that is required or a

demonstration o the rationality o Christian commitment is that the evi-

dence render Christianity at least as likely as not

Itrsquos sometimes said that the longest distance in the world is the distance

rom the head to the heart And so in part three Irsquoll try to illustrate how a

lie o Christian commitment is not just reasonable but worth desiring as





wellmdashsatisying both the head and the heart o do this Irsquoll tell the stories

o three exemplary individuals who took Jesus up on his invitation to ollow

him Te lives o these individuals were enriched and in turn enrichedcountless others in ways all o us would want to be true or our own lives

Dietrich Bonhoeffer Jean Vanier and Immaculeacutee Ilibagiza show how heroic

noble and beautiul a Christian lie can be Jesus said ldquoI came that they may

have lie and have it abundantlyrdquo (Jn 983089983088983089983088) Te abundance that so ofen

characterizes a devout Christian lie is not a matter o external goods or

material wealth but is internal and spiritual flowing rom a close personal

relationship with God Lie ocused on God offers comort inspiration anda peace that the vicissitudes o ortune need never take away

Te landscape o academic philosophy has changed significantly in the

last orty or fify years In the 983089983088s and rsquo1048631983088s it was taken or granted in many

philosophical circles that traditional Christian belie is unjustified and ir-

rational Tanks to the work o philosophers such as Alvin Plantinga

Eleonore Stump Peter van Inwagen and Richard Swinburne things have

changed

2

Recent decades have been a time o enormous productivity orphilosophers deending the rationality o Christian aith answering objec-

tions to Christian doctrines improving traditional arguments or the exis-

tence o God and ormulating new ones In the pages that ollow Irsquoll explore

some o the most important results o this body o work Irsquove written this

book though without presupposing that the reader has had ormal training

in philosophy or theology My hope is that it will be useul or anyone who

is interested in the evidence or God or the evidence or Christianity more

particularly I also hope that college students in philosophy and theology

courses will find the material worthwhile

Since this book is about evidence and reasons or religious belie itrsquos

worth addressing at the outset the question o why people believe what they

do on religious issues Isnrsquot religion a matter o the heart Does anyone really

believe on the basis o logic or argument Many committed Christians be-

lieve in God not primarily because o argument but because o experiencemdash

they have an intuitive sense o Godrsquos presence Perhaps not all the time but

certainly much o the time it just seems that God is there aware o what we

do and think Tis is how it is in my own case But what i you donrsquot happen

to have such an experience yoursel Or what i you have it occasionally but




8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

it is fleeting and open to serious doubt In those cases it makes good sense

to spend some time examining arguments and publicly available evidence

relating to questions about God and religion According to Christianityaith is a gif But or the person who does not have aith a careul look at

arguments or (and against) Christianity is a natural and reasonable step to

take Examination o evidence can also be worthwhile or the person who

does have aith but has doubts too In what ollows I give an extended

presentation o evidence or the truth o Christianity and a rigorous ar-

gument or the reasonableness o Christian commitment

While this book will ocus on the rational case or Christianity itrsquos im-portant to acknowledge that there are many actors involved in a decision

to commit to a Christian way o lie actors that go beyond impersonal

philosophical reasoning Onersquos upbringing onersquos experiences with indi-

vidual Christians the attitudes and views o onersquos closest riends and amily

onersquos emotional lie onersquos deep-seated hopes and ears and onersquos own par-

ticular way o viewing the worldmdashall o these come into play when one

encounters the message o Jesus I believe that the philosophical argumen-tation contained in this book will be helpul to many people but philo-

sophical argumentation is only one part o a larger picture When it comes

to religion logic may or may not be where one starts but itrsquos certainly not

where one should end Living a Christian lie is an act o the whole personmdash

mind and heart body and soul Still precisely because a Christian lie in-

volves the whole person there is a place or the mind and thus or reason

evidence and logic

Finally a ew words are called or regarding a concept that will play a

key role in what ollows probability Everyone has many belies but some

o those belies are held more confidently than others For example I be-

lieve that Irsquoll still be alive twenty-our hours rom now Irsquom not absolutely

certain I will be but nonetheless I do believe it I also believe Irsquoll be alive

twenty-our days rom now But Irsquom less confident in this belie than in my

belie about being alive twenty-our hours rom now How about twenty-

our years rom now In my case I wouldnrsquot say that I believe Irsquoll be alive

twenty-our years rom now I also donrsquot believe I wonrsquot be alive twenty-

our years rom nowmdashtherersquos just not enough inormation to orm a belie

one way or the other





Philosophers sometimes use the language o ldquolevels o confidencerdquo when

discussing belies and probability Levels o confidence come in a scale rom

zero to one (or 983088 percent to 983089983088983088 percent) o be absolutely certain o some-thing (eg 983089 + 983089 = 1048626) is to assign it a level o confidence o 983089 o be absolutely

certain a claim is alse (eg 983089 + 983089 = 9830891048631) is to assign it a level o confidence o

983088 In such a situation one could say that the probability o the claim is 983088

percent Terersquos just no chance that yoursquove gotten conused and ldquo983089 + 983089 = 9830891048631rdquo

is really true What about a proposition like ldquothis normal penny will come

up heads when I flip itrdquo Here itrsquos reasonable to assign equal levels o confi-

dence both to this claim and to the claim that the penny will come up tailsSo wersquod assign both claims a level o confidence o 9830881048629 and we could say that

the chance the coin will be heads is 1048629983088 percent How confident am I that this

six-sided die will show either a 983089 or a 1048626 when I roll it 104862710486271048627 percent You get

the idea

Te word probability will come up ofen in this book Irsquoll sometimes also

talk about how likely something is or what the chance is that such-and-such

is true Unless otherwise noted these are all just different ways to talk aboutthe same thing what philosophers call epistemic probability Roughly the

epistemic probability o a proposition (that is a statement or a claim) is the

level o confidence itrsquos reasonable to assign to that proposition3

Probability evidence moral heroism commitment all o these actors

will enter into our discussion o the reasonableness o Christian theism But

enough with introductory matters On to deeper waters




- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER

I983149983137983143983145983150983141 983156983144983137983156 983151983150983141 983140983137983161 you receive a phone call rom a abulously

wealthy individualmdashan eccentric billionaire known or spending his money

in odd and trivial ways He makes you a curious offer You are to determine

the religion you think most likely to be true and then practice that religion

devoutly attend religious services read the sacred scriptures o that religion

pray as that religion prescribes and live out the religionrsquos moral teachings

And he asks you to do this or the rest o your lie His part o the bargainwill be to place 983076983089983088 billion in an escrow account to be deliverable on your

death to whatever persons or causes you have named You can leave the

money to your amily or to a charitable organization or to scientific re-

search or any combination o the above whatever you think best I you

choose not to enter into the bargain he adds he will be using the money to

hire tens o thousands o artists to paint millions and millions o mediocre

oil paintings o his pet dog ScoutO course i this actually happened to you yoursquod be a mite skeptical But

imagine that afer due diligence you became convinced that this was no joke

And the billionaire clarified that you did not have to lie or otherwise mis-

represent yoursel in order to ulfill your side o the bargain I or example

you chose Christianity as the religion you thought most likely to be true you

would not have to recite the creed during the liturgy i you did not believe

it And your prayers could be conditional as in ldquoI yoursquore there God thank

you or this and please help me with thatrdquo An odd offer to be sure but also

in all probability the only chance yoursquoll ever have to direct 983076983089983088 billion o

wealth to persons or causes you care about So what would you do Would

you accept the offer




852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

Perhaps the act that the payoff is mere monetary gain gives you pause

Good or you to trade onersquos religious devotion or money would be base But

remember that you can name whatever beneficiary yoursquod like includingcharitable organizations that work with the poorest o the poor You can

trade your religious devotion or ood or the starving or education or

impoverished children in the developing world On board yet Say that the

billionaire adds that he doesnrsquot want to make this offer i you already happen

to be certain or almost certain that God does not exist His offer is only or

people who can say o some particular religion that there is at least a decent

chance that it is trueNow consider an alternative scenario Suppose that rom the get-go the

billionaire had offered you a different bargain the payoff is not a guaranteed

983076983089983088 billion but a 1048629983088 percent shot at 983076983089983088983088 billion Tat is suppose the bil-

lionairersquos side o the bargain is to put 983076983089983088983088 billion in an escrow account Ten

afer your death a air coin will be flipped I it comes up heads the entire

983076983089983088983088 billion will be given to your named beneficiary or beneficiaries I tails

then wersquoll have a lot o pictures o Scout Would you accept this offerHold on to that thought

Te overarching question o this book is the question o whether there is

good reason to make a serious commitment to living a Christian lie In part

two wersquoll take a tour o the evidence or Christianity But beore that the

question to ask first is ldquoHow much evidence would be required to make a

Christian commitment reasonablerdquo As Irsquoll try to show here in part one a

serious Christian commitment is recommended by reason even i there is

as little as a 1048629983088 percent chance that Christianity is true In other words i

you are willing to grant that Christianity is at least as likely as not then it is

rational or you to commit to living a devout Christian lie Indeed i yoursquore

in ordinary circumstances it is irrational or you not to1

P983137983155983139983137983148rsquo983155 W983137983143983141983154

Te argument Irsquoll give or this conclusion has its roots in the reflections o

seventeenth-century polymath Blaise Pascal Pascal made important contri-

butions in mathematics probability theory and physics His work on fluid

mechanics was instrumental in the later invention o the hydraulic press

and he himsel designed and built one o the first mechanical calculating




A Curious Of fer 852018852019

machines an early precursor to the digital computer (Interestingly the

principles used in the machine are still employed in many automobile

odometers) Pascal was an inventor an intellectual and a scientist He wasalso a deeply religious man At the age o thirty-one he had a powerul mys-

tical experience o God He described the experience in a note and stitched

the note into his coat keeping it close to him He evidently transerred the

paper note (together with a cleaner copy he put down on parchment) rom

coat to coat or the rest o his liemdasha servant ound them afer his death

Pascal suffered rom ill health dying at only thirty-nine In his final years

he was drafing a work o apologetics Te disorganized ragments that werehis notes or the project were published posthumously and quickly became

a classic Pascalrsquos Pensees (French or ldquothoughtsrdquo)2 In one o those ragments

Pascal gives an argument not or the theoretical conclusion that God exists

but or the practical conclusion that one should live a devout Christian lie

Te argument is addressed to those who arenrsquot sure whether Christianity is

true or alse and has come to be known as Pascalrsquos wager o be precise the

ragment contains the seeds o our different arguments in support o reli-gious commitment3 Te strongest version comes down to this It is rational

to seek a relationship with God and live a deeply Christian lie because there

is very much to gain and relatively little to lose In chapter two Irsquoll present a

strengthened and expanded version o this argument I Christianity is

indeed true then by committing to live a Christian lie one brings great joy

to God and all others in heaven raises the chance that one will be with God

orever raises the chance that one will help others attain union with God

expresses gratitude to God and becomes more aware o Godrsquos love and more

receptive to his help in the course o onersquos earthly lie On the other hand i

Christianity is alse the person who commits his or her lie to Jesusrsquo teachings

has still lived a worthwhile lie striving or moral excellence and experi-

encing the benefits o religious community (benefits that contemporary

sociological research reveals to be significant)

Pascalrsquos pragmatic argument in support o religious commitment has had

its share o critics Tus the amous English philosopher G E Moore ldquo[Con-

cerning Pascalrsquos claim that] in our state o doubt we should decide or that

belie which promises the greater reward I have nothing to say o it except

that it seems to me absolutely wickedrdquo4 Voltaire thought the wager was




852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

ldquosomewhat indecent and childish Te idea o gaming o losing or winning

is quite unsuitable to the dignity o the subjectrdquo5 In much the same vein an

earlier French critic had complained ldquoI lose patience listening to youtreating the highest o all matters and resting the most important truth in

the world the source o all truths on an idea so base and puerile on a com-

parison with a game o heads and tails more productive o mirth than per-

suasionrdquo6 More recently Christopher Hitchens sees Pascalrsquos theology as ldquonot

ar short o sordidrdquo7 remarking that ldquoPascal reminds me o the hypocrites

and rauds who abound in almudic Jewish rationalization Donrsquot do any

work on the Sabbath yoursel but pay someone else to do it You obeyed theletter o the law whorsquos countingrdquo8

Are these objections air Mathematician James Franklin suggests that

Pascalrsquos actual view is ofen replaced with an unflattering caricature o the

wager Franklin contrasts these two views

What Pascal said You have to choose whether to accept religion Tink o it

as a coin toss where you donrsquot know the outcome In this case i you losemdash

therersquos no Godmdashyou have not lost much But i you win there is an infinitepayoff So you should go to Mass and pray or aith [Franklinrsquos paraphrase]

Pascal caricatured Being base and greedy we want lots o goodies in this

lie and i possible the next So we are prepared to give up some pleasures

now on the off chance o a lot more later i our eye to the main chance makes

it look worth our while Since the loot on offer is infinite even a small chance

o raking it in makes it worth a try to grovel to any deity that might do what

we want9

Franklin makes a good point Pascalrsquos argument is ofen interpreted in

the worst possible light Still there are legitimate objections to consider

even when Pascalrsquos reasoning is interpreted airly It may be that it is im-

moral to allow any considerations about cost and benefit to affect what we

believe Perhaps a respect or truth requires that our belies be determined

by evidence and evidence alone Second we must consider whether the

costs o committing to God are simply too high And perhaps the most

common criticism is that Pascal illicitly assumed there were only two op-

tions to consider atheism and (Catholic) Christianity Te many gods ob-

jection as itrsquos ofen called contends that the multitude o possibly true reli-

gions makes trouble or Pascal





In chapters three and our Irsquoll elaborate on these and additional objec-

tions and respond to them But first we need to lay the basic argument

out on the table in more detail and with greater precision In order togive the strongest argument possible Irsquoll set aside the analysis o Pascalrsquos

text and instead develop a Pascalian argument in my own way10 Just or

the sake o having a handy title Irsquoll reer to my argument as ldquothe wagerrdquo

or even ldquoPascalrsquos wagerrdquo but this title should be taken merely as a re-

erence to its original inspiration not as a claim that I am providing a

verbatim reproduction o Pascalrsquos thought Indeed the wager Irsquoll present

takes advantage o cutting-edge work in philosophy and psychology andrecent sociological data unavailable to Pascal and it is thereore able to

greatly improve on his rather sketchy note Since itrsquos impossible to explain

the wager clearly without using some concepts rom decision theory

thatrsquos where wersquoll begin

D983141983139983145983155983145983151983150 T983144983141983151983154983161

Decision theory is about how to make good decisions in circumstances orisk or uncertainty Tree concepts are especially important states strat-

egies and outcomes A state is a possible way things might be (it will rain

today or it wonrsquot rain today) a strategy is a possible action the decider

might take (bring my umbrella or leave it at home) An outcome is a situ-

ation that results when a given strategy is taken and a certain state is actual

(eg the situation o being stuck in the rain without an umbrella) In order

to assess the suitability o the various available strategies the decision

maker attempts to orm a judgment about how valuable the various out-

comes are

For example suppose I offer you a bet I ask a trusted third party to flip

a air coin but not to tell us whether it lands heads or tails Ten I ask you

to decide either to play (that is participate in the bet) or pass I you choose

to play it will cost you one dollar but i you play and the coin has come up

heads you will get your dollar back and I will pay you two more dollars so

you will be up two dollars I you choose to play and the coin comes up tails

however I will keep your dollar so you will be down one dollar I on the

other hand you choose to pass no money will change hands either way

Would you take my bet




8520181048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

I bet you would Decision theory allows us to say precisely why the bet is

avorable or you Tere are two possible states or you to consider it could

be that the coin has come up heads and it could be that it has come up tailsAt the time you make your decision you donrsquot know which state is the actual

way things are but you do know that itrsquos one or the other Next there are

two possible strategies you can adopt play or pass Corresponding to the

our possible combinations o these states and strategies there are our pos-

sible situations that may resultmdashthese are the outcomes (983089) you play the

game and the coin is heads and so you win two dollars (1048626) you play the

game and the coin is tails so you lose one dollar (1048627) you pass and the coinis heads so you neither gain nor lose and (1048628) you pass and the coin is tails

so you neither gain nor lose In considering the question o what to do the

decision maker is invited to evaluate the outcomes Tat is the decision

maker attempts to estimate how valuable the various outcomes are In this

case the monetary value o outcome 983089 or you is two dollars or outcome 1048626

it is minus one dollar and or outcomes 1048627 and 1048628 zero In the case o most

actual decisions values other than money are at stake But starting with an

example using money is a good way to get an initial grasp o decision theory

Decision theorists commonly use a ldquodecision matrixrdquo to sum up inor-

mation relevant to a decision In our case we need a two-by-two matrix Te

columns correspond to the two possible states o the world and the rows

correspond to the two strategies you might take In the cells we write down

the valuations o the resulting outcomes

Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089

Should you play or pass I you only had one dollar to your name and

needed it to buy ood lest you starve then it might not be wise to play But

i yoursquore in ordinary circumstances and the loss o a dollar would be no great





loss then itrsquos rational to play Intuitively this is because yoursquove got an equal

chance o gaining and losing and i you gain you gain twice what you would

lose i you lost More precisely itrsquos because the expected value o playing ishigher than the expected value o passing Te expected value o a strategy

or a given game or bet is the average value per game that would accrue to

you i you played the game an infinite number o times and adopted that

strategy every time Here the expected value o the strategy ldquoplayrdquo can be

calculated by multiplying the probability o each state by the value o its

corresponding outcome and adding up the products So the expected value

o ldquoplayrdquo is equal to

(frac12) times (1048626) + (frac12) times (minus983089) = 1048629983088

I you were offered this bet a large number o times and you chose ldquoplayrdquo

every time yoursquod win an average o about fify cents per game Te expected

value o passing by contrast is

(frac12) times (983088) + (frac12) times (983088) = 983088

One reason playing is a good bet here is that the expected value o playing

(fify cents) is more than the expected value o passing (zero cents) Now as

it turns out itrsquos not always rational to take the strategy that maximizes onersquos

expected value11 But maximizing expected value is ofen a good way to

make decisions that involve risk

For the sake o clarity Irsquove illustrated the concepts o a state a strategy an

outcome and a decision matrix by giving an example with precise monetary

values or outcomes and with definite probabilities assigned to the possiblestates in question In real lie we usually donrsquot have precise probabilities to

assign to the states and much (perhaps most) o what we care about canrsquot

be valued in monetary terms Even in such cases however there are prin-

ciples that can guide our decision making

One such principle is at play when we say things like ldquoIrsquoll do X because it

canrsquot hurt and it might helprdquo or ldquoYou should do Y because therersquos nothing to

lose and maybe a lot to gainrdquo Suppose yoursquove borrowed your riendrsquos bicycleand have cycled to a store on the edge o town You put the lock around the

bike and a parking sign and as yoursquore lifing the key to lock up the bike it

occurs to you that this part o town is very sae and therersquos hardly any chance




8520181048632 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154

the bike will be stolen Besides when yoursquore inside the bike will be in your

view through the store window As you finish thinking this yoursquove also just

put the key into the lock So should you lock up the bike You donrsquot have anyprecise knowledge about the probability o thef in the circumstances and

while you could estimate the value o the bike therersquos no obvious way to

calculate the disvalue your riend would experience i your negligence got his

bike stolen Despite your lack o knowledge on those points though itrsquos clear

enough what you should do all you have to do to lock the bike is turn your

wrist A decision theorist could map out your situation like this

A thief will happen by A thief wont happen by

Lock Nothing bad happens Nothing bad happens

Dontlock

Bike stolen (probably) Nothing bad happens

Figure 983089 983090

I a motivated capable thie is in act coming along the strategy o locking

will result in a ar better outcome while i such a thie wonrsquot happen by locking

is no worse than unlocking (by the time o your decision itrsquos just so easy to

lock that therersquos no cost to locking) In the language o game theory we say

that the strategy o locking ldquoweakly dominatesrdquo the strategy o not locking

One strategy weakly dominates another i (and only i) there is at least one

possible state o the world on which the first strategy delivers a better outcome

than the second and there are no possible states o the world on which the

first strategy delivers a worse outcome than second In other words depending

on how things turn out adopting a weakly dominant strategy will either end

up yielding a better result than adopting the other strategy would have or it

will end up yielding a result that is just as good as what yoursquod have gotten with

the other strategy In a situation o uncertainty when we donrsquot know which o

several possible states o the world is the true one it is always better to adopt

a weakly dominant strategy instead o the strategy it dominates

In case yoursquore wondering one strategy strongly dominates another

strategy i and only i it delivers a better outcome on every possible state o





the world O course strongly dominant strategies are even more preerable

than weakly dominant strategies when you can get them

Decision theorists have other principles to help guide us in uncertaintyBut itrsquos worth noting that in some cases we donrsquot need a general principle

Sometimes we can just see what the smart thing to do is especially i wersquove

laid out the inormation in a decision matrix For example suppose again

that a trusted third party flips a coin or us but doesnrsquot say whether it lands

heads or tails Now imagine that I credibly make you this offer i you choose

to play and itrsquos heads then Irsquoll give you a million dollars while yoursquoll owe

me one dollar i itrsquos tails and yoursquove chosen to play I you choose to pass nomoney is exchanged either way We could represent this with the matrix

Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

Here we canrsquot say that playing weakly dominates passing because i the

coin is tails you do slightly better by passing But itrsquos one dollar wersquore talking

about Strictly speaking playing is not a weakly dominant strategy here but

itrsquos close enough Te payoff is so much greater i the coin is heads and the

loss is so little i itrsquos tails that itrsquos clearly rational to play Perhaps we couldstate a precise universally valid principle that would apply in this case and

tell us what to do but we donrsquot really need one we can just see what the

smart move is Tis insight will prove helpul later on

With these preliminaries behind us we can now turn to the wager A

consideration o this argument will allow us to see that Christian com-

mitment is recommended by reason so long as the available evidence makes

it at least as likely as not that the central claims o Christianity are true








T A K I N G

P A S C A L rsquo S

W A G E R


MI CH AE L ROTA




InterVarsity Press


ivpresscom

emailivpresscom



InterVarsity Press












ISBN 983097983095983096-983088-983096983091983088983096-983093983089983091983094-983090 (print)

ISBN 983097983095983096-983088-983096983091983088983096-983097983097983097983097-983097 (digital)







Michael Rota









P 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094 983089983093 983089983092 983089983091 983089983090 983089983089 983089983088 983097 983096 983095 983094 983093 983092 983091 983090 983089

Y 983091983093 983091983092 983091983091 983091983090 983091983089 983091983088 983090983097 983090983096 983090983095 983090983094 983090983093 983090983092 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094




CONTENTS



P983137983154983156 983089 U983150983139983141983154983156983137983145983150983156983161 983137983150983140 C983151983149983149983145983156983149983141983150983156

1048625 A Curious Oer 9830901048625






P983137983154983156 983090 E983158983145983140983141983150983139983141








1048625983090 Miracle or Myth 1048625983094983093

P983137983154983156 983091 S983137983161983145983150983143 Y983141983155 983156983151 G983151983140







Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095





INTRODUCTION



























852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154












































and predestination





as ollows




clusion o part two)


















852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154



















































changed

2






















8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















we want9




























D983141983139983145983155983145983151983150 T983144983141983151983154983161









comes are













8520181048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154






















Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































8520181048632 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154











Dontlock


Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

















T A K I N G

P A S C A L rsquo S

W A G E R


MI CH AE L ROTA




InterVarsity Press


ivpresscom

emailivpresscom



InterVarsity Press












ISBN 983097983095983096-983088-983096983091983088983096-983093983089983091983094-983090 (print)

ISBN 983097983095983096-983088-983096983091983088983096-983097983097983097983097-983097 (digital)







Michael Rota









P 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094 983089983093 983089983092 983089983091 983089983090 983089983089 983089983088 983097 983096 983095 983094 983093 983092 983091 983090 983089

Y 983091983093 983091983092 983091983091 983091983090 983091983089 983091983088 983090983097 983090983096 983090983095 983090983094 983090983093 983090983092 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094




CONTENTS



P983137983154983156 983089 U983150983139983141983154983156983137983145983150983156983161 983137983150983140 C983151983149983149983145983156983149983141983150983156

1048625 A Curious Oer 9830901048625






P983137983154983156 983090 E983158983145983140983141983150983139983141








1048625983090 Miracle or Myth 1048625983094983093

P983137983154983156 983091 S983137983161983145983150983143 Y983141983155 983156983151 G983151983140







Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095





INTRODUCTION



























852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154












































and predestination





as ollows




clusion o part two)


















852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154



















































changed

2






















8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















we want9




























D983141983139983145983155983145983151983150 T983144983141983151983154983161









comes are













8520181048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154






















Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































8520181048632 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154











Dontlock


Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

















InterVarsity Press


ivpresscom

emailivpresscom



InterVarsity Press












ISBN 983097983095983096-983088-983096983091983088983096-983093983089983091983094-983090 (print)

ISBN 983097983095983096-983088-983096983091983088983096-983097983097983097983097-983097 (digital)







Michael Rota









P 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094 983089983093 983089983092 983089983091 983089983090 983089983089 983089983088 983097 983096 983095 983094 983093 983092 983091 983090 983089

Y 983091983093 983091983092 983091983091 983091983090 983091983089 983091983088 983090983097 983090983096 983090983095 983090983094 983090983093 983090983092 983090983091 983090983090 983090983089 983090983088 983089983097 983089983096 983089983095 983089983094




CONTENTS



P983137983154983156 983089 U983150983139983141983154983156983137983145983150983156983161 983137983150983140 C983151983149983149983145983156983149983141983150983156

1048625 A Curious Oer 9830901048625






P983137983154983156 983090 E983158983145983140983141983150983139983141








1048625983090 Miracle or Myth 1048625983094983093

P983137983154983156 983091 S983137983161983145983150983143 Y983141983155 983156983151 G983151983140







Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095





INTRODUCTION



























852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154












































and predestination





as ollows




clusion o part two)


















852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154



















































changed

2






















8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















we want9




























D983141983139983145983155983145983151983150 T983144983141983151983154983161









comes are













8520181048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154






















Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































8520181048632 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154











Dontlock


Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

















CONTENTS



P983137983154983156 983089 U983150983139983141983154983156983137983145983150983156983161 983137983150983140 C983151983149983149983145983156983149983141983150983156

1048625 A Curious Oer 9830901048625






P983137983154983156 983090 E983158983145983140983141983150983139983141








1048625983090 Miracle or Myth 1048625983094983093

P983137983154983156 983091 S983137983161983145983150983143 Y983141983155 983156983151 G983151983140







Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095





INTRODUCTION



























852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154












































and predestination





as ollows




clusion o part two)


















852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154



















































changed

2






















8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















we want9




























D983141983139983145983155983145983151983150 T983144983141983151983154983161









comes are













8520181048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154






















Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































8520181048632 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154











Dontlock


Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

















Notes 983090983090983091

Author Index 983090983092983093

Subject Index 983090983092983095





INTRODUCTION



























852017852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154












































and predestination





as ollows




clusion o part two)


















852017852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154



















































changed

2






















8520171048630 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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Heads Tails

PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

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PlayOutcome 1

$1000000Outcome 2

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INTRODUCTION



























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and predestination





as ollows




clusion o part two)


















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changed

2






















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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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852018852020 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154





















we want9




























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comes are













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$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

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Dontlock


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$1000000Outcome 2

$-1

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$0Outcome 4

$0

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and predestination





as ollows




clusion o part two)


















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changed

2






















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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























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we want9




























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comes are













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$2Outcome 2

$-1

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$0Outcome 4

$0

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PlayOutcome 1

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$0Outcome 4

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and predestination





as ollows




clusion o part two)


















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changed

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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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$2Outcome 2

$-1

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$0Outcome 4

$0

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changed

2






















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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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$2Outcome 2

$-1

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$0Outcome 4

$0

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$0Outcome 4

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changed

2






















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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































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we want9




























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comes are













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$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

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$1000000Outcome 2

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$0Outcome 4

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evidence and logic



























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































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Dontlock


Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091





























the idea












- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























P983137983155983139983137983148rsquo983155 W983137983143983141983154













































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we want9




























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comes are













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PlayOutcome 1

$2Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089983089









































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Figure 983089 983090































Heads Tails

PlayOutcome 1

$1000000Outcome 2

$-1

PassOutcome 3

$0Outcome 4

$0

Figure 983089 983091

















- Part One -

UNCERTAINTY

and

COMMITMENT




- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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PlayOutcome 1

$2Outcome 2

$-1

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$0Outcome 4

$0

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Figure 983089 983090































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PlayOutcome 1

$1000000Outcome 2

$-1

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$0Outcome 4

$0

Figure 983089 983091

















- 983089 -

A CURIOUS OFFER



























852018852018 983137983147983145 983150983143 P983137983155983139983137 983148rsquo983155 W983137983143983141983154


























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we want9




























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comes are













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$2Outcome 2

$-1

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$0Outcome 4

$0

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Figure 983089 983090































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$1000000Outcome 2

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$0Outcome 4

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we want9




























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comes are













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we want9




























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comes are













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we want9




























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comes are













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$0Outcome 4

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$1000000Outcome 2

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comes are













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$0Outcome 4

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Documents

Taking Pascal's Wager By Michael Rota