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Probability, Choice,
and Reason
Probability, Choice,
and Reason

Leighton Vaughan Williams

First edition published 2022
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742

and by CRC Press

2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN

© 2022 Leighton Vaughan Williams

CRC Press is an imprint of Taylor & Francis Group, LLC

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Library of Congress Cataloging‑in‑Publication Data

A catalog record has been requested for this book

ISBN: 978-0-367-53893-4 (hbk)

ISBN: 978-0-367-53891-0 (pbk)
ISBN: 978-1-003-08361-0 (ebk)

DOI: 10.1201/9781003083610

Typeset in Palatino
by Deanta Global Publishing Services, Chennai, India
For Mum and Dad, and my wife, Julie.
Contents

Preface.................................................................................................................... xiii
Author Biography................................................................................................ xvii

1. Probability, Evidence, and Reason .............................................................. 1

1.1 Bayes’ Theorem: The Most Powerful Equation in the World..........1
1.1.1 Appendix................................................................................... 7
1.1.2 Exercise.......................................................................................9
1.1.3 Reading and Links.................................................................. 11
1.2 Bayes and the Taxi Problem................................................................ 13
1.2.1 Appendix................................................................................. 16
1.2.2 Exercise..................................................................................... 17
1.2.3 Reading and Links.................................................................. 18
1.3 Bayes and the Beetle............................................................................ 19
1.3.1 Appendix................................................................................. 20
1.3.2 Exercise..................................................................................... 20
1.3.3 Reading and Links.................................................................. 20
1.4 Bayes and the False Positives Problem.............................................. 20
1.4.1 Examples..................................................................................22
1.4.2 Appendix................................................................................. 23
1.4.2.1 Sensitivity and Specificity...................................... 23
1.4.2.2 Vaccine Efficacy....................................................... 24
1.4.3 Exercise..................................................................................... 25
1.4.4 Reading and Links.................................................................. 26
1.5 Bayes and the Bobby Smith Problem................................................ 27
1.5.1 Appendix................................................................................. 29
1.5.2 Exercise.....................................................................................30
1.5.3 Reading and Links.................................................................. 31
1.6 Bayes and the Broken Window.......................................................... 31
1.6.1 Appendix................................................................................. 33
1.6.2 Exercise..................................................................................... 33
1.7 The Bayesian Detective Problem.......................................................34
1.7.1 Epilogue................................................................................... 35
1.7.2 Exercise..................................................................................... 35
1.8 Bayesian Bus Problems........................................................................ 36
1.8.1 Exercise..................................................................................... 37
1.9 Bayes at the Theatre............................................................................. 37
1.9.1 Appendix................................................................................. 39
1.9.2 Exercise..................................................................................... 40
1.9.3 Reading and Links.................................................................. 40

vii
viii Contents

1.10 Bayes in the Courtroom...................................................................... 40

1.10.1 Exercise.....................................................................................44
1.10.2 Reading and Links..................................................................44

2. Probability Paradoxes................................................................................... 47
2.1 The Bertrand’s Box Paradox................................................................ 47
2.1.1 Exercise..................................................................................... 48
2.1.2 Reading and Links.................................................................. 49
2.2 The Monty Hall Problem.................................................................... 49
2.2.1 Appendix................................................................................. 52
2.2.1.1 Alternative Derivation............................................54
2.2.2 Exercise..................................................................................... 55
2.2.3 Reading and Links.................................................................. 55
2.3 The Three Prisoners Problem............................................................. 56
2.3.1 Exercise..................................................................................... 58
2.3.2 Reading and Links.................................................................. 58
2.4 The Deadly Doors Problem................................................................ 59
2.4.1 Exercise..................................................................................... 60
2.4.2 Reading and Links.................................................................. 60
2.5 Portia’s Challenge................................................................................. 61
2.5.1 Exercise..................................................................................... 62
2.5.2 Reading and Links.................................................................. 62
2.6 The Boy–Girl Paradox.......................................................................... 62
2.6.1 Appendix................................................................................. 67
2.6.2 Exercise..................................................................................... 68
2.6.3 Reading and Links.................................................................. 68
2.7 The Girl Named Florida Problem...................................................... 68
2.7.1 Appendix................................................................................. 71
2.7.2 Exercise..................................................................................... 72
2.7.3 Reading and Links.................................................................. 72
2.8 The Two Envelopes Problem.............................................................. 73
2.8.1 Exercise..................................................................................... 75
2.8.2 Reading and Links.................................................................. 75
2.9 The Birthday Problem......................................................................... 75
2.9.1 Exercise..................................................................................... 79
2.9.2 Reading and Links.................................................................. 79
2.10 The Inspection Paradox.......................................................................80
2.10.1 Exercise..................................................................................... 82
2.10.2 Reading and Links.................................................................. 82
2.11 Berkson’s Paradox................................................................................ 82
2.11.2 Exercise.....................................................................................84
2.11.3 Reading and Links.................................................................. 85
2.12 Simpson’s Paradox...............................................................................85
2.12.1 Exercise..................................................................................... 87
2.12.2 Reading and Links.................................................................. 88
Contents ix

2.13 The Will Rogers Phenomenon............................................................ 88

2.13.1 Exercise..................................................................................... 89
2.13.2 Reading and Links.................................................................. 89

3. Probability and Choice................................................................................. 91

3.1 Newcomb’s Paradox............................................................................. 91
3.1.1 Exercise..................................................................................... 93
3.1.2 Reading and Links.................................................................. 93
3.2 The Sleeping Beauty Problem............................................................ 93
3.2.1 Exercise..................................................................................... 96
3.2.2 Reading and Links.................................................................. 96
3.3 The God’s Coin Toss Problem............................................................ 97
3.3.1 Exercise................................................................................... 100
3.3.2 Reading and Links................................................................ 100
3.4 The Doomsday Argument................................................................ 100
3.4.1 Exercise................................................................................... 102
3.4.2 Reading and Links................................................................ 103
3.5 When Should You Stop Looking and Start Choosing?................. 103
3.5.1 Exercise................................................................................... 108
3.5.2 Reading and Links................................................................ 109
3.6 Why Do We Always Seem to End Up in the Slower Lane?......... 110
3.6.1 Exercise................................................................................... 111
3.6.2 Reading and Links................................................................ 111
3.7 Pascal’s Wager..................................................................................... 111
3.7.1 Exercise................................................................................... 113
3.7.2 Reading and Links................................................................ 113
3.8 The Keynesian Beauty Contest........................................................ 113
3.8.1 Exercise................................................................................... 115
3.8.2 Reading and Links................................................................ 115
3.9 Benford’s Law..................................................................................... 115
3.9.1 Exercise................................................................................... 118
3.9.2 Reading and Links................................................................ 118
3.10 Faking Randomness.......................................................................... 119
3.10.1 Exercise................................................................................... 120
3.10.2 Reading and Links................................................................ 120

4. Probability, Games, and Gambling......................................................... 123

4.1 The Chevalier’s Dice Problem.......................................................... 123
4.1.1 Exercise................................................................................... 128
4.1.2 Reading and Links................................................................ 129
4.2 The Pascal–Fermat “Problem of Points”......................................... 130
4.2.1 Appendix............................................................................... 131
4.2.2 Exercise................................................................................... 132
4.2.3 Reading and Links................................................................ 132
4.3 The Newton–Pepys Problem............................................................ 132
x Contents

4.3.1 Exercise................................................................................... 138

4.3.2 Reading and Links................................................................ 138
4.4 Staking to Reach a Target Sum........................................................ 138
4.4.1 Exercise................................................................................... 139
4.4.2 Reading and Links................................................................ 139
4.5 The Favourite-Longshot Bias............................................................ 140
4.5.1 Appendix............................................................................... 143
4.5.2 Exercise................................................................................... 146
4.5.3 Reading and Links................................................................ 147
4.6 The Poisson Distribution.................................................................. 148
4.6.1 Exercise................................................................................... 150
4.6.2 Reading and Links................................................................ 150
4.7 Card Counting.................................................................................... 150
4.7.1 Exercise................................................................................... 151
4.7.2 References and Links........................................................... 151
4.8 Can the Martingale Betting System Guarantee a Profit?............. 152
4.8.1 Appendix............................................................................... 154
4.8.2 Exercise................................................................................... 155
4.8.3 Reading and Links................................................................ 155
4.9 How Much Should We Bet When We Have the Edge?................. 156
4.9.1 Exercise................................................................................... 157
4.9.2 Reading and Links................................................................ 158
4.10 The Expected Value Paradox............................................................ 158
4.10.1 Exercise................................................................................... 161
4.10.2 Reading and Links................................................................ 161
4.11 Options, Spreads, and Wagers......................................................... 162
4.11.1 Appendix............................................................................... 166
4.11.2 Exercise................................................................................... 167
A. Buy Call Option.............................................................. 167
B. Buy Put Option................................................................ 168
C. Sell Call Option............................................................... 168
D. Sell Put Option................................................................ 169
4.11.3 Reading and Links................................................................ 170

5. Probability, Truth, and Reason................................................................. 171

5.1 Does Seeing a Blue Tennis Shoe Increase the Likelihood
That All Flamingos Are Pink?......................................................... 171
5.1.1 Exercise................................................................................... 173
5.1.2 Reading and Links................................................................ 173
5.2 The Simulated World Question........................................................ 174
5.2.1 Exercise................................................................................... 176
5.2.2 Reading and Links................................................................ 176
5.3 Quantum World Thought Experiments......................................... 176
5.3.1 Exercise................................................................................... 179
5.3.2 Reading and Links................................................................ 179
Contents xi

5.4 The Fine-Tuned Universe Puzzle..................................................... 179

5.4.1 Exercise................................................................................... 185
5.4.2 Reading and Links................................................................ 185
5.5 Occam’s Razor.................................................................................... 186
5.5.1 Exercise................................................................................... 190
5.5.2 Reading and Links................................................................ 190

6. Anomalies of Choice and Reason............................................................. 191

6.1 Efficiency and Inefficiency of Markets............................................ 191
6.1.1 Exercise................................................................................... 196
6.1.2 Reading and Links................................................................ 196
6.2 Curious and Classic Market Anomalies......................................... 197
6.2.1 Exercise................................................................................... 201
6.2.2 Reading and Links................................................................ 202
6.3 Ketchup Anomalies, Financial Puzzles, and Prospect Theory........ 203
6.3.1 Exercise................................................................................... 208
6.3.2 Reading and Links................................................................ 208
6.4 The Wisdom of Crowds.................................................................... 209
6.4.1 Exercise................................................................................... 213
6.4.2 Reading and Links................................................................ 214
6.5 Superforecasting................................................................................ 215
6.5.1 Exercise................................................................................... 218
6.5.2 Reading and Links................................................................ 218
6.6 Anomalies of Taxation...................................................................... 219
6.6.1 Exercise................................................................................... 220
6.6.2 Reading and Links................................................................ 220

7. Game Theory, Probability, and Practice.................................................223

7.1 Game Theory: Nash Equilibrium....................................................223
7.1.1 Exercise................................................................................... 229
7.1.2 Reading and Links................................................................ 229
7.2 Game Theory: Repeated Game Strategies...................................... 230
7.2.1 Exercise................................................................................... 231
7.2.2 Reading and Links................................................................ 232
7.3 Game Theory: Mixed Strategies...................................................... 232
7.3.1 Appendix...............................................................................234
7.3.2 Exercise................................................................................... 235
7.3.3 Reading and Links................................................................ 236

8. Further Ideas and Exercises....................................................................... 237

8.1 The Four Card Problem..................................................................... 237
8.1.1 Exercise................................................................................... 238
8.1.2 Reading and Links................................................................ 238
8.2 The Bell Boy Paradox......................................................................... 238
8.2.1 Exercise................................................................................... 239
xii Contents

8.3 Can a Number of Infinite Length Be Represented by

a Line of Finite Length?..................................................................... 239
8.3.1 Exercise................................................................................... 239
8.4 Does the Sum of All Positive Numbers Really Add Up
to a Negative Number?...................................................................... 239
8.4.1 Reading and Links................................................................ 241
8.5 Zeno’s Paradox.................................................................................... 241
8.5.1 Exercise................................................................................... 243
8.5.2 Reading and Links................................................................ 243
8.6 Cool Down Exercise........................................................................... 243
8.6.1 Exercise................................................................................... 243
8.6.2 Reading and Links................................................................ 243
Reading and References............................................................................... 244

Solutions to Exercises........................................................................................ 253

Index...................................................................................................................... 289
Preface

This book is designed as an invaluable resource for those studying the sci-
ences, social sciences, and humanities on a formal or informal basis, espe-
cially those with an interest in engaging with ideas rooted in chance and
probability and in the theory and application of choice and reason. Notably,
statistics and probability are topics that students often find difficult to get to
grips with, and this book fills and makes accessible a significant gap in this
area across a range of disciplines. These include economics, engineering,
finance, law, marketing, mathematics, medicine, psychology, and many oth-
ers. It will also appeal to those taking courses in probability and in statistics.
The target student audience includes university, college, and high school
students who wish to expand their reading, as well as teachers and lecturers
who want to liven up their courses while retaining academic rigour. More
generally, the book is designed with the intelligent and enquiring layperson
in mind, including anyone who wants to develop their skills to probe num-
bers and anyone who is interested in the many statistical and other para-
doxes that permeate our lives.
The underpinning of the book is that much of our thinking on a range of
subjects is flawed because we base much of our thinking on faulty intuition.
The content is primarily about the tools and framework of logical thought
that we can use to address and overcome these fundamental cognitive flaws.
By using the framework and tools of probability and statistics, we can
overcome these barriers to provide solutions to many real-world problems
and paradoxes. We show how to do this and find answers that are frequently
very contrary to what we might expect. Along the way, we venture into
diverse realms and thought experiments which challenge the way that most
of us see the world, and we explore the big questions of choice and reason.
The tools of so-called Bayesian reasoning run through important sections
of the book. The ideas extend well beyond a Bayesian framework, however,
and include several topics and anomalies that are attractive on their own
merits and from which we might learn broader lessons. The reader will also
explore ideas, concepts, and applications rooted in game theory.
A recurring theme at the heart of this book, however, is the conflict between
intuition and logic.
Imagine, for example, a bus that arrives every 30 minutes, on average, and
you arrive at the bus stop at some random time, with no idea when the last
bus left. How long can you expect to wait for the next bus to arrive? Half of
30 minutes, i.e. 15 minutes? Intuitively, that sounds right, but you’d be lucky
to wait only 15 minutes. It’s likely to be somewhat longer, and the laws of
probability and statistics show why.

xiii
xiv Preface

In medical trials, the success rate for a new drug is better than for an old
drug on each of the first two days of the trials. The new drug must, therefore,
have recorded a higher success rate than the old drug, judged over the entire
two days of the trials. Sounds right, but it’s not so. After the two days, the old
drug turns out to be more successful than the new drug even though it per-
formed worse on each of the first two days. Is this possible? It’s like saying
that a player performs better than another player in successive seasons but
performs worse overall. Can that happen? Yes. It can and does.
How many restaurants should you look at before starting to decide on a
place to eat? How many used cars should you pass on before you start look-
ing seriously for one? How many potential partners should you consider
before looking for the special one? In each case, we can derive the answer
from a simple formula.
A doctor performs a test on all her patients for a virus. The test she gives
them is 99% accurate, in the sense that 99% of people who have the virus test
positive, and 99% of the healthy people test negative. Now the question is:
If the patient tests positive, what is the chance the doctor should give to the
patient having the virus? The intuitive answer is 99%, but that is likely to be
a gross over-estimate of the true probability.
You meet a man at a sales convention, who mentions his two children, one
of whom, you learn, is a boy. You never found out anything about the other
child. What should be your best estimate of the probability that the other
child is a girl? It’s not a half, as you might think intuitively. If you had met the
same man in different circumstances, accompanied by his son, now what is
the probability that the man’s other child is a girl? Isn’t it the same as before?
In fact, it’s quite different. It does matter in estimating the chance of the other
child being a girl that you bumped into the young boy instead of being told
about him. It matters that his name was Barrington, and it would be slightly
different if it were Bob.
You turn up to watch your local team play football. There are 22 players
on the pitch, plus the referee. What’s the chance that two or more of them
share a birthday? Well, there are 365 days in the year and only 23 people on
the pitch, so the chance is likely to be slim, you might think. In fact, it’s more
likely than not that at least two of those on the pitch share the same birthday.
But the referee is unlikely to be one of them.
Can we improve our forecasts of football match outcomes by studying the
rate of fatalities from horse kicks of Prussian cavalry officers? Yes, we can.
Can we devise a game where you auction a dollar and be pretty much
guaranteed to turn a profit on the deal? There is a way to do this.
You have arranged to meet a stranger on a particular day for an important
appointment, but you forgot to name the time and place, and neither of you
have the contact details of the other. Where and when should you turn up?
You need to double your remaining money to pay off a pressing debt, and
you decide to take to the casino tables. If you don’t double up tonight, you are
Preface xv

doomed to a dusty demise. What staking plan should you adopt to maximise
your chances of survival? The answer may be surprising.
It’s possible to win at Blackjack by counting cards, memorising what cards
have already been dealt. You need a good memory for that, don’t you? You
don’t.
Choose a number between 0 and 100. You win a prize if your number is
equal or closest to two thirds of the average number chosen by all other par-
ticipants. What number should you choose?
Select a newspaper or magazine with a lot of numbers about naturally
occurring phenomena, such as the populations of different countries or the
heights of mountains. Now circle the numbers. Would you expect a very big
difference between the numbers starting with a 1, 2, 3, 4, 5, 6, 7, 8, or 9? Yes,
you would. And that fact can help identify fraudsters.
As a prize for winning a competition, you’re offered a chance to open a
gold, silver or lead casket, in one of which the host has placed a cheque for
£10,000. The others are empty. You choose the gold casket and the silver cas-
ket is opened. It is empty. You are generously offered a chance to swap to a
different casket before the reveal. Should you take the offer? The solution is
counter-intuitive.
The penalty-taker must decide which way to shoot. The goalkeeper must
decide which way, and whether, to dive. How can they use game theory
to maximise their chances of success? The answer involves thinking both
inside and outside the box.
Can we profit on the stock market by waiting till Halloween or by invest-
ing on a cold, overcast day?
Are professional golfers more successful when putting for par than for
birdie?
Does seeing a blue tennis shoe increase the likelihood that all flamingos
are pink?
Do we live in a simulation or is this world the real thing?
How long can we expect humanity as we know it to survive?
We ask and seek to resolve these and many more questions involving prob-
ability, choice, and reason. Not least, we solve the greatest mystery of them
all – why we always seem to end up in the slower lane.
Exercises, references, and links are provided for those wishing to cross-
reference or to probe further, and many of the chapters contain a technical
appendix. Solutions to the exercises are provided at the end of the book.
Author Biography

Leighton Vaughan Williams, BSc (Econ), PhD, FHEA, is Professor of

Economics and Finance at Nottingham Business School, Nottingham Trent
University, as well as Director of the Betting Research Unit and of the
Political Forecasting Unit. He has researched and published extensively in
the areas of probability, risk, and choice under uncertainty and given expert
witness evidence before national and international courts of law and select
committees of the House of Commons and House of Lords. He has served
as a senior adviser to the UK government and teaches undergraduates and
postgraduates how to apply Bayesian methods, and the tools of probability
and statistics, to real-world problems and paradoxes.

xvii
1
Probability, Evidence, and Reason

This chapter introduces Bayes’ Theorem, named in honour of the Reverend

Thomas Bayes. Bayes’ Theorem offers a way to update the probability of a
hypothesis being true, given some new evidence, using a simple but very
powerful mathematical equation. Bayesian updating is in this way a solu-
tion to the problem of how to combine pre-existing (prior) beliefs with new
evidence. We also introduce the Bayes Factor, which is the ratio of the likeli-
hood of one hypothesis to the likelihood of another. It is essentially a mea-
sure of which hypothesis better explains the world, given the evidence. We
examine the Prosecutor’s Fallacy and Laplace’s Rule of Succession and show
some applications of Bayesian reasoning. These include the classic taxi prob-
lem, the beetle problem and the false positives problem, the latter taking us
into the realms of health and medicine. We also look at the application of
Bayesian reasoning in the real-world courtroom. Stylised examples include
the Bayesian detective, the Bobby Smith problem, and Bayes at the theatre.

1.1 Bayes’ Theorem: The Most Powerful Equation in the World

How should we change our beliefs about the world when we encounter
new data or information? A theorem bearing the name of Thomas Bayes, an
eighteenth-century clergyman, is central to the way we should answer this
question.
The original presentation of the Reverend Thomas Bayes’ work, “An Essay
toward Solving a Problem in the Doctrine of Chances”, was given in 1763,
after Bayes’ death, to the Royal Society, by Bayes’ friend and confidant,
Richard Price.
In explaining Bayes’ work, Price proposed, as a thought experiment, the
example of a person who enters the world and sees the sun rise for the first
time. Perhaps he has spent his entire life entombed in a dark cave. As this
person has had no previous opportunity to observe dawn, he is not able to
decide whether this is a typical or unusual occurrence. It might even be a
unique event. Every day that he sees the same thing happen, the degree of
confidence he assigns to this being a permanent aspect of nature increases.
His estimate of the probability that the sun will rise again tomorrow as it

DOI: 10.1201/9781003083610-1 1
2 Probability, Choice, and Reason

did yesterday and the day before, and so on, gradually approaches but never
quite reaches 100%.
The Bayesian viewpoint is just like that, the idea that we learn about the
world and everything in it through a process of gradually updating our
beliefs. In this way, we edge closer to the truth as we obtain more data, more
information, more evidence.
The Bayes Business School, formerly City University of London’s business
school, explained their choice of name in similar terms: “Bayes’ theorem sug-
gests that we get closer to the truth by constantly updating our beliefs in pro-
portion to the weight of new evidence. It is this idea … that is the motivation
behind adopting this name” (Significance, June 2021, p. 3).
As such, the perspective of Reverend Bayes differs from that of philoso-
pher David Hume. For Hume, assumptions about the future, such as that the
sun will rise again, cannot be rationally justified based simply on the past
because no law exists that the future will always resemble the past. Bayes
instead sees reason as a practical matter, to which we can apply the laws of
probability in a systematic way.
To Bayes, therefore, we step ever nearer to the truth based on new evi-
dence and the proper application of the laws of probability. This is called
Bayesian reasoning. According to this approach, we can see probability as
a bridge between ignorance and knowledge. Bayes’ Theorem is, in this way,
concerned with conditional probability. It tells us the probability, or updates
the probability, that a theory or hypothesis is correct, given that we observe
some new evidence. A particularly good thing about Bayesian reasoning is
that the mathematics of it is so straightforward.
At its heart, then, Bayes’ Theorem allows us to use all the information
available to us. Our beliefs, our judgments, our subjective opinions, what
we have already learned from the previous body of knowledge to which we
have had access. We can incorporate this in updating our estimate of the
probability that a hypothesis is true. As such, we can be explicit and open
about the uncertainty in our data and our beliefs. The problem with implicit
reasoning, or intuition, is that our intuition is often wrong and subject to
systematic biases. Instead, we should be trained to think in a Bayesian way
about the world.
Often the conclusions generated by the application of Bayes’ Theorem will
challenge intuition. This is because the world is, in many ways, a counter-
intuitive place. Accepting that fact is the first step towards mastering life’s
logical maze.
Intuition also often lets us down because our in-built judgment of the
weight that we should attach to new evidence tends to be skewed relative to
pre-existing evidence.
New evidence also tends to colour our perception of the pre-existing evi-
dence. Moreover, we tend to see evidence that is consistent with something
being true as evidence that it is in fact true. Bayes’ Theorem is the map that
helps guide us through this maze.
Probability, Evidence, and Reason 3

Essentially, though, Bayes’ Theorem is just an algebraic expression with

three known variables and one unknown. Yet this simple formula is the
foundation stone of that bridge between ignorance and knowledge, which
can lead to critical predictive insights. Bayesian reasoning allows us to use
this formula to update the probability that a theory or hypothesis is true
when some new evidence comes to light.
There are three things a Bayesian needs to estimate.

1. A Bayesian’s first task is to assign a starting point probability to

a hypothesis being true before some new evidence arises. This is
known as the “prior” probability. Let’s assign the letter “a” to this.
2. A Bayesian’s second task is to estimate the probability that the new evi-
dence would have arisen if the hypothesis was correct. This is some-
times known as the “likelihood”. Let’s assign the letter “b” to this.
3. A Bayesian’s third task is to estimate the probability that the new
evidence would have arisen if the hypothesis was false. Let’s assign
the letter “c” to this.

Based on these three probability estimates, Bayes’ Theorem offers a way to

calculate the revised probability of the hypothesis being true, given the new
evidence. A notable point is that the equation is true as a matter of logic. The
result it produces will be as accurate as the values inputted into the equation.
The formula is also so straightforward that it can be jotted down on the back
of a hand.
We can represent the formula for Bayes’ Theorem as follows:
Updated (posterior) probability given new evidence = ab/ [ab + c (1 − a)].
Bayesian updating is thus a straightforward solution to the problem of
how to combine pre-existing (prior) beliefs with new evidence. The solution
is essentially to combine the probabilities. To do this properly, we use Bayes’
Theorem. It is of particular use when we have a conditional probability of
two events, and we are interested in the reverse conditional probability. For
example, when we have P (A given B) – the probability of A given B – and
want to find P (B given A) – the probability of B given A.
Looked at from another angle, Bayes’ Theorem allows us to calculate the
probability of certain events occurring conditional on other events that may
occur.
The probability that event B happens given that event A has occurred is
given by the formula:

P (BIA) = P (AIB) . P (B)/P (A)

This is the conditional probability of event B given event A, which is calcu-

lated by multiplying the conditional probability of event A given event B by
the probability of event B, divided by the probability of event A.
4 Probability, Choice, and Reason

This idea can also be applied to beliefs. So, P (BIE) can be understood as the
degree of belief, B, given evidence, E. P (B) is our prior degree of belief before
we encountered evidence, E. Employing Bayes’ Theorem allows us to con-
vert our prior belief into a posterior belief. When new evidence is observed,
we can perform the same calculation again, this time our previous posterior
belief becoming our next prior belief. And so on. As McGrayne (2011, preface)
puts it, “by updating our initial belief about something with objective new
information, we get a new and improved belief. To its adherents, this is an
elegant statement about learning from experience”.
More generally, the probability that a hypothesis is true, P (H), given new
evidence, P (E), is written as: P (H I E).

Now, P(H I E) = P (E I H) . P(H)/P (E).

Where P (E) is the probability of the evidence.

P(E) = P(E I H) × P(H) + P (E I 1 - H) . P (1 - H)/P(E).

P (1 − H) can also be written as P (H’).

In words, probability of the evidence = probability of the evidence given
that the hypothesis is true times probability the hypothesis is true plus prob-
ability of the evidence given that the hypothesis is not true times probability
that the hypothesis is not true.

In a, b, c notation, a = P (H), b = P (E I H), c = P (E I 1 − H) or P (E I H’).

The problem with P (E) is that it’s often difficult to calculate it in many real-
world cases. In such cases, it may sometimes be preferable to use the Bayes
Factor, which is a formula for comparing the plausibility of one hypothesis
with another.

The Bayes Factor

Bayes’ Theorem states that:

P (H I E) = P (H) . P (E I H) /P (E).

But it’s not always clear how to measure P (E), the probability of the evidence.
An alternative approach which doesn’t require knowledge of P (E) is by
using the proportional form of Bayes’ Theorem.
The proportional form of Bayes’ Theorem sees the posterior probability of
a hypothesis, P (H I E), as proportional to the prior probability, P (H), multi-
plied by the likelihood, P (E I H).
From this we derive a ratio of how well each of our hypotheses explains
the evidence we have observed.
Probability, Evidence, and Reason 5

The formula is: P (H1​) × P​(E I​H1) /​ [P (​ H2) ×

​ P (​E I H​2)], where P (H1) is
the probability that hypothesis 1 is true and P (H2) is the probability that
hypothesis 2 is true.
If the ratio is five, for example, this means that H1 explains the evidence
five times as well as H2, and vice versa if the ratio is 1/5.
Using this formula, assume that P (H1) = P (H2), i.e. our prior belief in each
hypothesis is the same. In this case, P (H1) / P (H2) = 1.
This leaves: P (E I H1) / P (E I H2). This ratio is known as the Bayes Factor.
Bayes Factor = P (E I H1) / P (E I H2).
The Bayes Factor is thus the ratio of the likelihood of one hypothesis to the
likelihood of the other. It is essentially a measure of which hypothesis better
explains the world, given the evidence.
This reasoning assumes, however, that the prior probability of each
hypothesis is the same. This may not be true. Before observing the evidence,
one hypothesis may be considered more likely than another. We should also
consider, therefore, the ratio of prior probabilities: P (H1) / P (H2).
When used in conjunction with the Bayes Factor, this ratio is commonly
termed the “prior odds”, written as O (H1), which is a measure of how
likely H1 is relative to the competing hypothesis before we observe the new
evidence.
From these ratios, we can calculate the posterior (or updated) odds.

Posterior odds = O (H1) . P (E I H1) / P (E I H2).

The posterior odds is a measure of how many times better our hypothesis
explains the evidence compared to a competing hypothesis.
Take as an example a hypothesis that a machine is a perfect Coin Toss
Predictor, in that it can unfailingly calculate how a coin will land face up as
soon as it is thrown. You toss a fair coin a series of times.
If the Predictor is perfect, it will always calculate correctly, so P (E I H1) =
1, i.e. the probability of calling it correctly (E) given that the hypothesis, H1
(it is a perfect Predictor), is true = 1.
The alternative hypothesis, H2, is that the Predictor is simply guessing. In
this case, P (E I H2) = 0.5 from one toss of the coin.
Say you toss the coin five times, and the Predictor calls it correctly five times.
In this case, the probability of doing this by chanc​e = 0​.5 × ​0.5 ×​0.5 ×
​ 0.5​ ×
0.​5 = (​0.5)5 = 1/32. We can multiply the probabilities as each coin toss is an
independent event.

Here, Bayes Factor = 1 / (0.5)5 = 32.

If the original (prior) probability we attach to the Predictor being genuine is

equal to it being a guessing machine, then:
​ O (​H I E​) = O​(H) ​. P (​E I H​1) / ​P (E I​ H2)​= 1 ​× 32 ​= 32.​This
Posterior odds =
means that the hypothesis that the machine is a perfect Coin Toss Predictor
6 Probability, Choice, and Reason

explains the evidence we have witnessed 32 times better than the alternative
hypothesis, that it is a guessing machine.
If, on the other hand, the prior probability we assign to the coin being a
genuine perfect Predictor compared to a guesser is 1/64, then the posterior
odds = 1/64 × 32 = 1/2. Now, we believe that it is twice as likely that the
machine is a guessing machine than a perfect Coin Toss Predictor.
How big do the posterior odds have to be to prove convincing? To some
extent that depends on what you are using them for. If it’s to help resolve
a casual disagreement among friends, a small positive number might be
enough. If your life depends on getting it right, you might prefer that num-
ber to end in quite a few zeros!

Prosecutor’s Fallacy
The Prosecutor’s Fallacy is to represent P (HIE) as an equivalent to P (EIH).
In fact, P (HIE) = P (EIH) P (H) / P (E) … Bayes’ Theorem. Therefore, P
(HIE) only equals P (EIH) when P (H) = P (E), i.e. P (H) / P (E) = 1. Bayes’
Theorem can be expanded to: P (HIE) = P (EIH) P (H) / [P (EIH) P (H) + P
(EIH’) P (H’)]

Laplace’s Rule of Succession and Bayesian “Priors”

Laplace’s Rule of Succession, named after Pierre-Simon Laplace, is a rule-of-
thumb way of calculating how likely it is that something that has happened
before will happen again. The method is to count how many times it has
happened in the past plus one (successes, S + 1) and divide that by the num-
ber of opportunities for it to have happened, plus two (trials, T + 2). For a per-
son emerging from a dark cave into the world for the first time and watching
the sun rise seven times, for example, the estimate that it will rise again the
next day is: (S + 1​ ) / (​ T + 2​ ) = (​ 7 + 1​ ) / (​ 7 + 2​ ) = 8​ /9 =​88.9​%. Every time it rises
again makes it even more likely that the pattern will repeat so that by the end
of a year, the estimated probability goes up to (365 + 1) / (365 + 2) = 99.7%.
And so on. The 1 and 2 in the Laplace equation, (S + 1) / (T + 2), represents a
version of the Bayesian “prior”. The 1 and 2 can be replaced by any numbers
in the same proportion, such as 5 and 10 or 10 and 20, depending upon how
anchored we are to our prior beliefs or understanding of the world.
Larger numbers (e.g. S + 10, T + 20) lead to slower updating in response to
new evidence. So (S + 10) / (T + 20) after seven days updates to a probability
of (7 + 10) / (7 + 20) = 17/27 = 63.0%, compared to 88.9% for (S + 1) / (T + 2).
So, smaller numbers indicate that we are more open to quickly updat-
ing our beliefs based on new evidence. In other words, updating takes
place more quickly and readily with smaller numbers in the Laplace
equation.
In conclusion, the core contributions of Bayesian analysis to our under-
standing of the world are threefold.
Probability, Evidence, and Reason 7

1. Bayes’ Theorem makes clear the importance of not just new evidence
but also the (prior) probability that the hypothesis was true before
the arrival of the new evidence. This prior probability may in com-
mon intuition be given too little (or too much) weight relative to the
latest evidence. Bayes’ Theorem makes the assigned prior probabil-
ity explicit and shows how much weight to attach to it.
2. Bayes’ Theorem allows us a way to update the probability that a
hypothesis is true. It does so by combining the prior probability with
the probability that the new evidence would arise if the hypothesis
is true and the probability that it would arise if the hypothesis is
false.
3. Bayes’ Theorem shows that the probability that a hypothesis (H) is
true given the evidence (E) is not equal to the probability of the evi-
dence arising given that the hypothesis is true, except in limiting
circumstances. Specifically, P (H given E) does not equal P (E given
H) except when P (H) = P (E).

1.1.1 Appendix
Bayes’ Theorem consists of three variables.

• a is the prior probability of the hypothesis being true (the probability

we attach before the arrival of new evidence). In traditional notation,
this is represented as P (H).
• b is the probability that the new evidence would arise if the hypoth-
esis is true. In traditional notation, we represent this as P (EIH). We
use the notation P (AIB) to represent the probability of A given B.
• c is the probability the new evidence would arise if the hypothesis is
not true. In traditional notation, we represent this as P (EIH’). H’ is
the notation for H not being true.

(1 − a) is, therefore, the prior probability that the hypothesis is not true. In
traditional notation, we represent this as P (H’) or 1 − P (H), i.e. one minus
the probability that the hypothesis is true.
Using the a, b, c notation, the probability that a hypothesis is true given
some new evidence (“posterior probability”) = ab/ [ab + c (1 − a)].

Deriving Bayes’ Theorem

If A and B are two independent events, we can write the conditional prob-
ability that A occurs, given that B has occurred, as P (AIB), which is the prob-
ability of A given B.
Now, the formula for Bayes’ Theorem states that: P (AIB) = P (BIA) . P
(A) / P (B).
8 Probability, Choice, and Reason

This follows from the definition of conditional probability: P (AIB) = P (A

and B) / P (B).
To see how Bayes’ Theorem is derived, note that P (AIB) = P (A and B) / P
(B) and also that P (BIA) = P (B and A) / P (A).
But the event “A and B” is the same as “B and A”. Dividing one equation by
the other, we have P (AIB) / P (BIA) = P (A) / P (B).
Now, multiply both sides by P (BIA).

P ( A I B ) = P ( BIA ) × P ( A ) /P ( B ) ¼ Bayes’ Theorem

More Formal Derivation

The conditional probability of H given E is conventionally represented as
P (HIE). It can be defined as the probability that the hypothesis, H, occurs,
given that the evidence, E, occurs.
Now, the probability that both H and E occur, P (H∩E), is the conditional
probability of H occurring given E multiplied by the probability that E
occurs.

P ( H Ç E ) = P ( HIE ) × P ( E )

Similarly,

P ( E Ç H ) = P ( EIH ) . P ( H )

Now,

P (H Ç E) = P (E Ç H)

So:

P ( HIE ) P ( E ) = P ( EIH ) P ( H )

Dividing both sides by P (H) (which we take to be non-zero), the result

follows:

P (HIE) P (E)/ P (H) = P(EIH)

So, P (HIE) = P (EIH) P (H) / P (E) … Bayes’ Theorem.

Expanding the denominator, P (E) = P (EIH) P (H) + P (EIH’) P (H’), where
P (H’) represents the probability that the hypothesis is not true, i.e. P (H’) =
1 − P (H).

P ( HIE ) = P ( EIH ) P ( H ) éëP ( EIH ) P ( H ) + P (EIH¢) P (H¢)ùû ¼ Bayes’ Theorem

Probability, Evidence, and Reason 9

This can also be stated as:

P ( HIE ) = P ( EIH ) P ( H ) éëP ( EIH ) P ( H ) + P (EIH¢) P (1 - H)ùû ¼ Bayes’ Theorem

Intuitive Presentation
Bayes’ Theorem can be derived from the equation P (HIE) . P (E) = P (H) . P
(EIH).
The intuition underlying this equation is that both sides are alternative
ways of looking at the same thing. It is the combined probability of observ-
ing the evidence relating to a hypothesis and the probability that the hypoth-
esis is true, P (H and E).

So, P (HIE) . P (E) = P (H) . P (EIH).

Dividing both sides of the equation by P (E),

P ( HIE ) = P ( H ) × P ( EIH ) /P ( E ) ¼ Bayes’ Theorem

P ( E ) = P ( EIH ) × P ( H ) + P(EIH¢) × P(H¢)

P ( HIE ) = P ( H ) × P ( EIH ) éëP ( H ) × P ( EIH ) + P(EIH¢) × P(H¢)ùû ¼ Bayes’ Theorem

This can also be stated as:

P ( HIE ) = P ( EIH ) P ( H ) éëP ( EIH ) P ( H ) + P(EIH¢) P(1 - H)ùû ¼ Bayes’ Theorem

This is equivalent to the formula:

Posterior probability = ab/[ab + c (1 - a)], where a = P (H);

b = P (EIH); c = P (EIH¢).

1.1.2 Exercise

Question a.
Write the Bayesian equation (using a, b, and c) for deriving the poste-
rior (updated) probability of a hypothesis being true after the arrival
of new evidence. Explain what a, b, and c represent.
Question b.
If P (H) is the probability that a hypothesis is true before some new
evidence (E), what is the updated (or posterior) probability after the
10 Probability, Choice, and Reason

new evidence? Use the terms P (H), P (EIH), P (HIE), P (H’), and P
(EIH’) to construct the Bayesian equation.
Question c.
How do the terms used in Question b relate to a, b, and c in the
Bayesian formula referred to in Question a?
Question d.
1. Is the probability that a hypothesis is true, given the evidence,
P (HIE), equal to the probability of the evidence, given that the
hypothesis is true, P (EIH)? In other words, does P (HIE) = P
(EIH)?
2. Is the probability of feeling warm given that you are out in the
sun equal to the probability of being out in the sun given that
you are feeling warm?
Question e.
For a person emerging from a dark cave into the world for the first
time and watching the sun rise seven times, the estimate that it will
rise again is 88.9%, if we use a Bayesian “prior” of 1, 2. Calculate
the updated probabilities that the sun will rise again if we use a
Bayesian “prior” of 5, 10? What is the significance of using a 5, 10
prior compared to a 1, 2 prior?
Question f.
Uncle Austin and Uncle Idris each present you with a die. One is fair,
and one is biased. The fair die (A) lands on all numbers (1–6) with
equal probability. The biased die (B) lands on 6 with a 50% chance
and each of the other numbers (1–5) with an equal 10% chance each.
Now, choose one of the two dice at random. You can’t tell by
inspection whether it is the fair or the biased die. You now roll the
die, and it lands on 6. What is the probability that the die you rolled
is the biased die?
Answer guide: state the hypothesis to be that you chose the biased die.
What is P (H)? What is the probability that the die is biased before
the evidence that the die landed on a 6?
What is P (EIH)? Note that the evidence is that the die landed on a 6.
What is P (EIH’), i.e. the probability that you would throw a 6 if the
die was not biased?
What is P (HIE)?
Alternatively, you can use the formula: ab / [(ab + c (1 − a)].
Question g.
Auntie Beatrice and Auntie Kit each present you with a coin. One
of these is a fair coin, and the other is weighted. The fair coin (Coin
1) lands on heads and tails with equal likelihood, the weighted coin
(Coin 2) lands on heads with a 75% chance.
Probability, Evidence, and Reason 11

Now, choose one coin. You can’t tell by inspection whether it is

the fair or the weighted coin. You select a coin and toss it, and it
lands on heads. What is the probability that you tossed Coin 2 (the
weighted coin)?
Question h.
You own a pair of dice (one blue and one red) and are told that your
colleague can always guess how they will land as soon as they leave
your hands. They are your own dice so you know there are no tricks.
You throw the dice in the air and he calls out: blue will show 5 and
red will show 6. That’s exactly how they end up.
In terms of the probability that your colleague is a genuine perfect
Dice Predictor compared to a guesser, what is the Bayes Factor? What
are the posterior odds if you were originally perfectly split between
believing in his powers and believing he was a guesser? What are the
posterior odds if you originally believed that he was 100 times more
likely to be a guesser than a genuine perfect Dice Predictor?
Question i.
Every time you meet to play tennis, your friend, May, tosses a coin to
determine who serves first. You know that she prefers to serve first
and are a tiny bit, but only a tiny bit, suspicious that the coin tosses
are not fair and are designed to land heads, so that she can choose
to serve first. You play once a week and over the course of 12 weeks,
she always calls heads and the coin lands heads every time. What
is the Bayes Factor for the hypothesis that she is cheating compared
to a hypothesis that the coin tosses were fair? What if, despite your
hypothesis, you are almost (though not totally) certain that your
friend is playing fair, so you assign a probability of 1 in 1,000 that
she would cheat? What are the posterior odds now for the hypoth-
esis that the coin toss was rigged to always land heads? What do the
posterior odds represent here?

1.1.3 Reading and Links

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Bayes, T. and Price, R. 1763. An Essay towards solving a Problem in the Doctrine
of Chances. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter
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Ellerton, P. 2014. Why facts alone don’t change minds in our public debates. The
Conversation. 13 May. https​:/​/th​​econv​​ersat​​ion​.c​​om​/wh​​y​-fac​​ts​-al​​one​-d​​ont​-c​​
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Probability, Evidence, and Reason 13

Bayes Theorem. 3Blue1Brown. 22 December 2019. YouTube. https://youtu​.be​/

HZGCoVF3YvM
Bayes: How One Equation Changed the Way I Think. Galef, J. 4 June 2013. YouTube.
https://youtu​.be​/za7RqnT7CM0
Bayes’ Theorem – The Simplest Case. Bazett, T. 19 November 2017. YouTube. https://
youtu​.be​/XQoLVI31ZfQ
Bayes’ Theorem/Law, Part 1. patrickJMT.3 January 2013. YouTube. https://youtu​.be​/
E4rlJ82CUZI
Bayes’ Theorem/Law, Part 2. patrickJMT. 3 January 2013. YouTube. https://youtu​.be​/
zh1E8cGoV7k
FRM: Bayes’ Formula. Bionic Turtle. 9 January 2008. YouTube. https://youtu​.be​/
pPTLK5hFGnQ
Prior Indifference Fallacy. Fuggetta, M. 24 January, 2014. YouTube. https://youtu​.be​/
PkcUM3Mr​_F4
The Bayesian Trap. Veritasium. 2017. 5 April 2017.. YouTube. https://youtu​.be​/
R13BD8qKeTg

1.2 Bayes and the Taxi Problem

To help explain how we can apply Bayes’ Theorem in practice, let’s start
with the classic Bayesian Taxi Problem. It goes something like this. New
Amsterdam has 1,000 taxis – 850 are blue and 150 are green. One of these
taxis knocks down a pedestrian and then is driven away without stopping.
We have no prior reason to believe that the driver of a blue taxi is more likely
to have knocked down the pedestrian than of a green taxi, or vice versa.
There is one independent witness, however, who did see the event. The wit-
ness says the colour of the taxi was green. The witness is given a rigorous
observation test, which recreates as carefully as possible the event in ques-
tion, and her judgment has proved correct 80% of the time.
So what is the probability that the taxi was green?
The intuitive answer is in the region of 80%, as the only evidence is that
of the witness, and the test of her powers of observation shows that she is
right 80% of the time. That is not the Bayesian approach, however, which
is to consider the evidence in the light of the baseline, or prior, probability
that the taxi was green before the witness evidence came to light. We can
derive the prior probability from an identification of the proportion of taxis
in New Amsterdam that are green, which is 15% (of the 1,000 taxis, 150 are
green).
Now, the (posterior) probability that a hypothesis is true after obtain-
ing new evidence, according to the a, b, c formula of Bayes’ Theorem, is
equal to:

ab / [ab + c(1 - a)]

14 Probability, Choice, and Reason

In this case, the hypothesis is that the taxi that knocked down the pedestrian
was green, where:

• a is the prior probability, i.e. the probability that a hypothesis is true

before the new evidence arises. This is 0.15 (15%) because 15% of the
taxis in New Amsterdam are green.
• b is the probability the new evidence would arise if the hypothesis is
true. This is 0.8 (80%). There is an 80% chance that the witness would
say the taxi was green if it was indeed green.
• c is the probability the new evidence would arise if the hypothesis is
false. This is 0.2 (20%). There is a 20% chance that the witness would
be wrong and identify the taxi as green if it was blue.

Inserting these numbers into the formula, ab / [ab + c (1 − a)], gives:

Posterior probability = 0.15 ´ 0.8 / [0.15 ´ 0.8 + 0.2 (1 - 0.15)] = 0.41 = 41%.

In other words, the actual probability that the taxi that knocked down the
pedestrian was green is not 80% (despite the witness evidence) but about half
of that. The baseline probability is important. A common error is to place too
much weight on new evidence about an event (the judgment of the witness)
and too little on the general frequency of that event (in this case, represented
by the proportion of green cabs in the taxi population).
If new evidence subsequently arises, Bayesians are not content to leave the
probabilities alone. Say, for example, that a second witness appears and is
also given the observation test, revealing a reliability score of 90%. Again,
we have no reason to doubt the integrity of this second witness. A Bayesian
now inserts that number (0.9) into Bayes’ formula (b = 0.9) so that c (the prob-
ability that the witness is mistaken) = 0.1. The new baseline (or prior) prob-
ability, a, is no longer 0.15, as it was before the first witness appeared, but 0.41
(the probability incorporating the evidence of the first witness). In this sense,
yesterday’s posterior probabilities are today’s prior probabilities.
Inserting into Bayes’ Theorem, the new posterior probability = 0.86 = 86%.
This is the new baseline probability underpinning any further new evidence
which might arise.
There are three critical illustrative cases of the Bayesian Taxi Problem
which bear highlighting. The first is a scenario where the new witness scores
50% on the observation test. Here is a case where intuition and Bayes’ for-
mula converge. A witness who is right only half the time is also wrong half
the time, and so any evidence they give is worthless. Bayes’ Theorem tells us
that this is indeed so, as the posterior probability ends up being equal to the
prior probability.
The second illustrative case is where a new witness is 100% reliable about
the colour of the taxi. In this case, b = 1 and c = 0. Intuition tells us that the
Probability, Evidence, and Reason 15

evidence of such a witness solves the case. If the infallible witness says the
taxi was green, it was green. Bayes’ Theorem agrees.
Now for the third illustrative case. If the new witness scores 0% on the
observation test, this indicates that they always identify the wrong colour for
the taxi. If they say it is green, it is not green. So the chance (posterior prob-
ability) that the cab is green if they say so is zero, which accords with Bayes’
Theorem.
More generally, information that informs us that a witness is usually
wrong is valuable, as it can be reversed to beneficial effect. A witness who
always identifies a green taxi as blue and vice versa, and is 100% consistent in
doing so, yields us reliable information by merely reversing their designated
colour.
So if the witness says the taxi is blue, we can now identify the taxi as defi-
nitely being green. This now converges on the second illustrative case.
Similarly, a witness who is, say, right only 25% of the time in identifying
the colour of the taxi in the observation test also yields us valuable informa-
tion. By reversing the defined colour, this produces a 75% reliability score,
which can be inserted accordingly into Bayes’ Theorem to update the prob-
ability that the taxi that knocked down the pedestrian was green. In other
words, a witness who is 25% reliable and identifies the cab as green is equiv-
alent to the witness being 75% reliable in determining the taxi as blue, and
vice versa.
The only observation evidence that is worthless, therefore, is evidence that
could have been produced by the flip of a coin.
The Bayesian Taxi Problem is an instance of what is known as the Base
Rate Fallacy. This occurs when we undervalue prior information when mak-
ing a judgement as to how likely something is. If presented with general
(base rate) information and specific information (pertaining only to a par-
ticular case), the fallacy arises from a tendency to focus on the latter at the
expense of the former. For example, if someone is an avid book enthusiast,
we might think it more likely that they work in a bookshop or a library than
as, say, a nurse. There are, however, many more nurses than librarians and
bookshop assistants. Our mistake is not to take sufficient account of the base
rate numbers for each occupation.
And the conclusion to the case? CCTV evidence was later produced in
court, which was able to identify the taxi and the driver conclusively. The
pedestrian never regained consciousness. The driver of what transpired to
be a blue taxi told the jury that the pedestrian unexpectedly stepped out and
lightly brushed against the passenger side door. He thought at the time that
it was a minor incident and was completely unaware that the victim had
slipped and hit his head awkwardly. This account was rejected by the jury,
who accepted the prosecution’s contention that the driver had acted with
premeditation and malicious intent. They based their decision on their view
that a driver who was so motivated would indeed have driven off. It was all
they needed to reach their unanimous verdict of first-degree murder.
16 Probability, Choice, and Reason

James Parker, a 29-year-old long-time resident of New Amsterdam, of pre-

vious good character, with no prior convictions or any known motive for the
crime, is currently serving a sentence of life in a maximum-security prison
with no possibility of parole. No member of the jury, it turned out, had ever
heard of the Reverend Thomas Bayes or the Prosecutor’s Fallacy.

1.2.1 Appendix
In the original taxi problem scenario:

a = 0.15 (15% of taxis are green)

b = 0.8 (the witness is correct 80% of the time)
c = 0.2 (the witness is wrong 20% of the time)

Inserting these numbers into the formula gives:

Posterior proba​bilit​y = (​0.15 ×

​ 0.8​) / (​0.15 ×
​ 0.8​+ 0.​2 × 0​.85) =
​ 0.1​2 / (​0.12 ​
+ 0.1​7) = 4​ 1% (rounded to the nearest per cent).

This is the new baseline probability underpinning any new evidence which
might arise.
If new evidence subsequently arises, this should be used to update the new
baseline probability of 0.41.
Say, for example, that a new witness is correct 90% of the time (wrong 10%
of the time). New posterior proba​bilit​y = 0​.41 ×​ 0.9 ​/ (0.​41 × ​0.9 +​0.1 ×
​ 0.5​9) =
​0.369​/ (0​.369 ​+ 0.0​59) =​86% (rounded to the nearest per cent). This is also
the new baseline probability underpinning any further new evidence which
might arise.
Solution to the three illustrative cases of the Bayesian Taxi Problem:

1. A scenario where the new witness scores 50% on the observation

test. In terms of the equation, b = 0.5 and c = 0.5.

Posterior probability = ab / éëab + c ( 1 - a ) ùû = 0.5a / éë0.5a + 0.5 ( 1 - a ) ùû

= 0.5a / ( 0.5 + 0.5a - 0.5a ) = 0.5a / 0.5 = a

So when b and c both equal 0.5 in regard to new evidence, this evi-
dence has no impact on the probability of the hypothesis being tested
being true. The posterior probability equals the prior probability. In
this case, the evidence of the witness can be discounted.
2. The second illustrative case is where a new witness is 100% accurate
about the colour of the taxi. In this case, b = 1 and c = 0. Intuition tells
us that the evidence of such a witness solves the case. If the infallible
Probability, Evidence, and Reason 17

witness says the taxi was green, it was green. Bayes’ formula agrees.
Inserting b = 1 and c = 0 into the formula gives:

ab / éëab + c ( 1 - a ) ùû = a / ( a + 0 ) = a / a = 1

So the new (posterior) probability that the taxi is green = 1.

3. This leads directly to the third illustrative case. If the new witness
scores 0% on the observation test, this indicates that they always
identify the wrong colour for the taxi. If they say it is green, it is not
green. So the chance (posterior probability) that the taxi is green if
they say so is zero. This accords with the formula.

ab / éëab + c ( 1 - a ) ùû = 0 / éë0 + ( 1 - a ) ùû , assuming a is not equal to 1 = 0

Note that a cannot equal 1 if c equals 1. It would represent a logical con-

tradiction, implying within the context of the illustration that every taxi is
green but a witness who is always wrong says that the taxi is green.

1.2.2 Exercise
For the purpose of this exercise, use the a, b, c method to derive the solutions.

Question a.
New Amsterdam has 1,000 taxis, and 800 of them are yellow and
200 are white. The driver of one of these taxis knocks down a
pedestrian and drives away. There is no prior reason to believe
that the driver of a yellow taxi is more likely to have knocked
down the pedestrian than of a white taxi, or vice versa. There is
one witness, however, who saw the event and says the colour of
the cab was white.
The witness, Reverend Latimer Williams, is given a well-respected
observation test and is right 80% of the time.
What is our best estimate now of the probability that the taxi was
white?
Question b.
What if a second witness now comes forward?
We determine that the probability that this witness is correct when
identifying the colour of the taxi as 70%.
The witness, Mr. Henry Morris, says the colour of the taxi was white.
What is the new posterior (updated) probability that the taxi that
knocked down the pedestrian is white?
18 Probability, Choice, and Reason

Question c.
What if a third witness now comes forward?
We determine that the probability that this witness is correct when
identifying the colour of the taxi as 50%.
The witness, Mr. Edmund Coss, says the colour of the taxi was white.
What is the new posterior (updated) probability that the taxi that
knocked down the pedestrian is white?
Question d.
A witness, Mr. Smith, is correct 50% of the time.
A witness, Mr. Jones, is correct 100% of the time.
A witness, Mr. Evans, gets it wrong 100% of the time.
Which of the three witnesses is the most useful/least useful to
investigators?

1.2.3 Reading and Links

Bedwell, M. 2015. Slow thinking and deep learning: Tversky and Kahneman’s cabs.
Global Journal of Human-Social Science, 15, 12. file:​///C:​/User​s/epa​3will​ilv/D​
ownlo​ads/1​634-1​-1640​-1-10​-2016​0225%​20(1)​.pdf
Gesmann, M. 2014. Hit and run. Think Bayes! 29 July. https​:/​/ma​​gesbl​​og​.co​​m​/pos​​t​/
201​​4​- 07-​​29​-hi​​t​-and​​-run​-​​think​​-baye​​s/
plusadmin. 1997. Solution to the taxi problem. + plus magazine. 1 May. https​:/​/pl​​us​
.ma​​ths​.o​​rg​/co​​ntent​​/solu​​tion-​​ta​xi-​​probl​​em
plusadmin. 1997. Solution to the taxi problem revisited. + plus magazine. 1 May.
https​:/​/pl​​us​.ma​​ths​.o​​rg​/co​​ntent​​/solu​​tion-​​taxi-​​probl​​​em​-re​​visit​​ed
Salop, S.C. 1987. Evaluating uncertain evidence with Sir Thomas Bayes: A note for
teachers. Economic Perspectives, 1, 1, Summer, 155–160. https​:/​/pu​​bs​.ae​​aweb.​​org​/
d​​oi​/pd​​f​/10.​​1257/​​​jep​.1​​.1​.15​5
Talwalkar, P. 2013. Mind your decisions. The taxi-cab problem. 5 September. https​:/​/
mi​​ndyou​​rdeci​​sions​​.com/​​blog/​​2013/​​09​/05​​/the-​​taxi-​​c​ab​-p​​roble​​m/
Taxi Cab Problem, Exercise in Value and Information. Putt, B. 12 April 2020. YouTube
link. https​:/​/ww​​w​.you​​tube.​​com​/w​​atch?​​v​=ySw​​​wiji0​​B7o
The Decision Lab. 2021. Why do we rely on event-specific information over statis-
tics? Base Rate Fallacy, explained. https​:/​/th​​edeci​​sionl​​ab​.co​​m​/bia​​ses​/b​​ase​-r​​a​
te​-f​​allac​​y/
Tversky, A., and Kahneman, D. 1982. Evidential impact of base rates. In Kahneman,
D., Slovic, P. and Tversky, A. (eds.), Judgment under Uncertainty: Heuristics and
Biases. https​:/​/ww​​w​.cam​​bridg​​e​.org​​/core​​/book​​s​/jud​​gment​​-unde​​r​-unc​​ertai​​nty​/
e​​viden​​tial-​​impac​​t​-of-​​base-​​rates​​/CC35​​C9E39​​​07270​​85713​​C4E6D​​0D1D4​​633
Woodcock, S. 2017. Base rate fallacy. In: Paradoxes of probability and other statistical
strangeness. UTS, 5 April. http:​/​/new​​sroom​​.uts.​​edu​.a​​u​/new​​s​/201​​7​/04/​​parad​​
oxes-​​proba​​bilit​​y​-and​​-othe​​r​-sta​​t​isti​​cal​-s​​trang​​eness​
Woolley, R. 2016. Do I call or fold? How Bayes’ theorem can help navigate Poker’s
uncertainty, part 1. 15 February. https​:/​/ww​​w​.pok​​ernew​​s​.com​​/stra​​tegy/​​call-​​or​-
fo​​ld​-ba​​yes​-t​​heore​​m​-pok​​er​-un​​​certa​​inty-​​24077​​.htm
Probability, Evidence, and Reason 19

Woolley, R. (2016). Do I call or fold? How Bayes’ Theorem can help navigate Poker’s
uncertainty, part 2. 22 February.
https​:/​/ww​​w​.pok​​ernew​​s​.com​​/stra​​tegy/​​call-​​or​-fo​​ld​-ba​​yes​-t​​heore​​m​-pok​​er​-un​​cer​ta​​
inty-​​2​-241​​33​.ht​m
Base rate fallacy. Wikipedia. https​:/​/en​​.wiki​​pedia​​.org/​​wiki/​​Base_​​rate_​​​falla​​cy
Base Rate Fallacy. Yang, C. 22 March 2017. YouTube. https://youtu​.be​/Fs8cs0gUjGY
Counting Carefully – The Base Rate Fallacy. Simple Scientist. 20 May 2013. YouTube.
https://youtu​.be​/VeQXXzEJQrg
Know Your Bias: Base Rate Neglect. Deciderata. 25 July 2016. YouTube. https://youtu​.
be​/YuURK​_q2NR8

1.3 Bayes and the Beetle

A nature lover spots what might be a rare type of beetle due to the pattern on
its back. In the rare category, 98% have the pattern. In the common category,
only 5% have the pattern. The rare category accounts for only 0.1% of the
population. How likely is the beetle to be rare?
Intuition might suggest that we have come across a rare insect when
observing the unusual pattern. This is because there is a very high chance of
a rare beetle having this pattern and a low chance of a common beetle hav-
ing this pattern. Bayes’ Theorem tells us something entirely different.
To calculate just how likely the beetle is to be rare given that we see the
pattern on its back, we apply Bayes’ Theorem.

Posterior probability = ab / éëab + c ( 1 - a ) ùû

Where a is the prior probability of the hypothesis (beetle is rare) being true.
b is the probability we observe the pattern, and the beetle is rare (hypothesis
is true). c is the probability we observe the pattern, and the beetle is not rare
(hypothesis is false).
In this case, a = 0.001 (0.1%); b = 0.98 (98%); c = 0.05 (5%).
So, updated proba​bilit​y = a​b / [​ab + ​c (1 ​− a)]​= 0.​0192.​So there is just a
1.92% chance that the beetle is rare when the distinctive pattern is spotted
on its back.
Why the counter-intuitive result? Few beetles are rare, so it would take a
lot more evidence than observing the rare pattern to alter the prior expecta-
tion that the beetle is not rare.
So the probability that the beetle is rare (the hypothesis) given that we
observe the distinctive pattern (the evidence) is 1.92%. What is the chance,
however, that we will observe the distinctive pattern if the beetle is rare? In
other words, what is the chance of observing the evidence (the pattern) if the
hypothesis (the beetle is rare) is correct? That is 98%.
20 Probability, Choice, and Reason

To believe these two things are the same is a common mistake known as
the Inverse (or Prosecutor’s) Fallacy. In this instance, it is to believe that the
chance of observing the pattern given that the beetle is rare (98%) is the same
as the chance that the beetle is rare given the observation of the pattern (the
actual probability that the beetle is rare, which is 1.92%).

1.3.1 Appendix
We can also solve the beetle problem using the traditional notation version
of Bayes’ Theorem.

P ( HIE ) = P ( EIH ) . P ( H ) éëP ( EIH ) . P ( H ) + P(EIH¢) . P(H¢)ùû

In this case, P (H) = 0.001 (0.1%); P (EIH) = 0.98 (98%); P (EIH’) = 0.05 (5%).
So, P (HI​E) = ​0.98 ​× 0.0​01 / ​[0.98​ × 0.​001 +​ 0.05​ × 0.​999)]​ = 0.​00098​ / 0.​00098​
+ 0.​04995​= 0.​00098​/ 0.​05093​= 0.​0192.​So there is just a 1.92% chance that the
beetle is rare when the entomologist spots the distinctive pattern on its back.
Note also that P (HIE) = 0.0192, while P (EIH) = 0.98.
The Prosecutor’s Fallacy is to conflate these two expressions.

1.3.2 Exercise
A nature lover spots what might be a rare category of beetle, due to the pat-
tern on its back. In the rare category, 95% have the pattern. In the common
category, only 2% have the pattern. The rare category accounts for only 1% of
the population. How likely is the beetle to be rare?
In solving this question, what are a, b, and c?
Solve again, using traditional notation, in the case where 5% (instead of
2%) of those in the common category have the pattern.

1.3.3 Reading and Links

AgileKiwi. 2011. Bayes’ theorem demystified. 30 December. http:​/​/www​​.agil​​ekiwi​​.
com/​​off​-t​​opic/​​bayes​​-theo​​rem​-d​​​emyst​​ified​/
CS201. Bayes’ theorem. Excerpts from Wikipedia. https​:/​/ma​​thcs.​​clark​​u​.edu​/​~jma​​
gee​/c​​s201/​​slide​​s​/Bay​​e​sThe​​orem.​​pdf
Thompson, J. 2011. Bayes’ theorem. 20 November. https​:/​/ww​​w​.jef​​freyt​​homps​​on​.or​​g​/
blo​​g​/201​​1​/11/​​20​/b​a​​yes​-t​​heore​​m/

1.4 Bayes and the False Positives Problem

A patient goes to see the doctor. The doctor performs a test on all her patients
for a virus, estimating in advance of the test that 1% of those who visit the
Probability, Evidence, and Reason 21

surgery have the virus. The test is 99% accurate, in the sense that 99% of
people with the virus test positive, and 99% of those who do not have the
virus test negative.
Let us say that the first patient tests positive. What is the chance that the
patient has the virus?
The intuitive answer is 99%, as the test is 99% accurate. But is that right?
The information we are given relates to the probability of testing positive
given that you have the virus. What we want to know, however, is the probabil-
ity of having the virus given that you test positive. This is a crucial difference.
Common intuition conflates these two probabilities, but they are very dif-
ferent. If the test is 99% accurate, this means that 99% of those with the virus
test positive. But this is not the same thing as saying that 99% of patients
who test positive have the virus. This is another example of the “Inverse
Fallacy” or “Prosecutor’s Fallacy”. In fact, those two probabilities can diverge
markedly.
So what is the probability you have the virus if you test positive, given that
the test is 99% accurate? To answer this, we can use Bayes’ Theorem.
The probability that a hypothesis is true after obtaining new evidence,
according to the a, b, c formula of Bayes’ Theorem, is equal to: ab / [ab + c
(1 − a)], where:

• a is the prior probability, i.e. the probability that a hypothesis is true

before you see the new evidence. Before the new evidence (the test),
this chance is estimated at 1 in 100 (0.01), as 1% of the people who
visit the surgery have the virus. So, a = 0.01.
• b is the probability of the new evidence if the hypothesis is true. The
probability of the new evidence (the positive result on the test) if the
hypothesis is true (the patient has the virus) is 99% since the test is
99% accurate. So, b = 0.99.
• c is the probability of the new evidence if the hypothesis is false. The
probability of the new evidence (the positive result on the test) if the
hypothesis is false (the patient does not have the virus) is just 1%
because the test is 99% accurate. So, c = 0.01.

Using Bayes’ Theorem, the updated (posterior) proba​bilit​y = a​b / [​ab +

​c (1 −​ a)]​= 1/​2.
So there is a 50% chance that the patient has the virus if testing positive.
It is basically a competition between how rare the virus is and how rarely
the test is wrong. In this case, there is a 1 in 100 chance that you have the
virus before taking the test, and the test is wrong one time in 100. These two
probabilities are equal, so the chance that you have the virus when testing
positive is 1 in 2, despite the test being 99% accurate.
But what if the patient is showing symptoms of the virus before being
tested?
Another random document with
no related content on Scribd:
8
In speaking of Doris, Herodotus says: Ή δὲ χώρη αὕτη ἐστὶ
μητρόπολις Δωριέων τῶν ἐν Πελοποννήσῳ. That the land had a
Dorian population in the fifth century b.c. is undoubtedly the case;
but its claim to be metropolis of the Dorians of the south was in all
probability set up by the Spartan authorities, as affording a
convenient pretext for interference in Greek affairs north of Isthmus.
It is probable that this corner of Greece, of which the Malian plain
was the centre, contained patches of various peoples which had in
different ages traversed the peninsula, or which had been driven into
its mountain fastnesses by the passage of invaders:⁠—Dorians,
Œtæans, Trachinians, etc., were probably such remains of larger
tribes.
9
Herodotus seems from his language to assume (viii. 31) that the
whole army had come south by the Dorian route. That is, either a
mistake; or, more probably, the impression his language gives is due
to a mere omission. Few details are given of any part of the route of
the army.
0
The position at Delphi, from a military point of view, is by no means
weak, provided Amphissa be occupied, and the great pass from the
north be thus closed. Under those circumstances, unless the
assailant is in a position to land troops at the head of the Krissæan
gulf, the only line of attack is along this easily defensible path from
the west. It is imaginable that Xerxes, knowing it to be an open town,
under-estimated the difficulty of its capture.
Pogon is an almost land-locked harbour between the island of
Kalauria and the mainland.
2

A Comparison of the Lists of Vessels at Artemisium and

Salamis respectively.
T. = trireme; P. = pentekonters.
Artemisium. Salamis.
 Artemisium. Salamis.
{127 T.
Athenians (some Platæans in crews at
53 T. later} 180 T.
Artemisium)
180 T.
Corinthians 40 T. 40 T.
Megareans 20 T. 20 T.
Chalkidians in Athenian ships 20 T. 20 T.
Æginetans 18 T. 30 T.
Sikyonians 12 T. 15 T.
Lacedæmonians 10 T. 16 T.
Epidaurians 8 T. 10 T.
Eretrians 7 T. 7 T.
Trœzenians 5 T. 5 T.
Styreans 2 T. 2 T.
Keians 2 T.; 2 P. 2 T.; 2 P.
Opuntian Locrians 7 P. —
Hermionians — 3 T.
Ambrakiots — 7 T.
Leukadians — 3 T.
Naxians — 4 T.
Kythnians — 1 T.; 1 P.
Krotonians — 1 T.
Malians — 2 P.
Siphnians — 1 P.
Seriphians — 1 P.
 Artemisium. Salamis.
366 T.;
324 T.; 9 P.
7 P.
Æschylus gives 310 as the number of the Greek fleet. Valuable as
is the testimony of the poet with regard to those incidents in the
battle which he observed as an eye-witness, his evidence on the dry
question of numbers is not likely to be exact.
3
She did, indeed, send sixty vessels, to observe, so said the patriot
Greeks, how the war went, but not with any intention of taking part
therein. The Corcyræans’ own excuse for their non-participation was
that their fleet had been unable to round Malea.
4
The mistake may be that of a manuscript copyist; but such mistakes
are so common in the text of Herodotus, that they afford strong
ground for supposing that the historian was, like the men of his time,
inaccurate in numerical calculations. The mistake may be in the
Paus. ii. 29. 5. detailed list. Pausanias implies that the Æginetan
contingent was superior in numbers to that of the
Corinthian, that is to say, more than forty. If the number were forty-
two, the total given by Herodotus would be correct; and it is
noticeable in this reference that he himself, in speaking of the
H. viii. 46. number of ships which Ægina supplied, says: “Of the
islanders the Æginetans supplied thirty; they had indeed
other ships manned; but with these they were guarding their own
country; but with the thirty best sailers they fought at Salamis.”
5
By Professor J. W. Bury.
6
Macan, Herod, iv., v., vi., “Athens and Ægina.”
7
Note on the Reference to Siris in Themistocles’ Speech.—The
reference to Siris inevitably suggests that this reported passage in
Themistocles’ speech is an invention of later date arising from the
colonization of Thurii in or about 443. The rapid growth of Athenian
trade in the earlier part of the fifth century, and its peculiar
development along the western route, render it possible, however,
 that an idea of settlement on or near the deserted city of Sybaris
may have been long anterior to the actual settlement, and may have
been mooted even before 480. If Plutarch is to be believed,
Themistocles had direct relations with Corcyra, and gave the name
of Sybaris to one, and the name of Italia to the other of his daughters
(Plut. Them. 32).
8
H. viii. 74. τέλος δὲ ἐξερράγη ἐς τὸ μέσον. Cf. also Diod. xi. 16, ad
fin.
9
This Council of War must have been held on the morning of the day
preceding the battle. It lasted, in all probability, several hours, and, if
so, this would indicate the afternoon as the time at which Xerxes
received the message of Themistocles. On this point, then, the
indications in the narrative of Æschylus and Herodotus are in
agreement.
0
Plut. Them. also mentions the same name; but the testimony is
probably dependent on that of Herodotus.
There is a curious triangular concord at this point in the history.
Diodorus says that the Egyptian contingent was sent to block the
strait towards the Megarid (xi. 17).
Plutarch says 200 vessels were sent to close the passage round
Salamis (Them. 12).
Herodotus mentions that the Egyptian contingent numbered 200
(vii. 89).
2
This would account for the fact implied by Æsch. Pers. 400: the two
fleets when they started their movement were not in sight of one
another, though, very shortly after the movement began, the Persian
fleet was visible to the Greeks. The latter would first catch sight of it
after it rounded the Kynosura promontory and the island.
3
Cf. Arist. 8, where the revocation is said to have taken place τρίτῳ
ἔτει after the sentence.
4
Cf. Stein’s brief note on the translation of the words στὰς ἐπὶ τὸ
συνέδριον in H. viii. 79.
5
In so far as I know, this last very important point was first raised by
Prof. J. B. Bury in an article in the Classical Review on “Aristides at
Salamis.”
6
This is Professor Bury’s suggestion. It is open to the objection that
Herodotus expressly mentions the arrival of this vessel (H. viii. 83)
immediately before the battle began. But this objection is not by any
means insuperable. It is much more probable, under the
circumstances, that Herodotus made a mistake as to the time of its
arrival, than that it managed at the time he mentions to force its way
through the blockading fleets at either end of the strait.
7
It would seem as if it were a description of this movement, taken
from his notes on, or sources of information for, the details of the
battle, which Herodotus has used by mistake in describing the
movement of the Persian fleet during the night. He has, of course,
intensely confused the original description by reading into it what he
knew to be the object of that night-movement—the surrounding of
the Greek fleet by blocking the issues both to east and west of it; but,
eliminating this motive from his description, it is possible to see that
in its original form it must have resembled very closely the
description of the advance of the Persian fleet which has been drawn
from the details which Æschylus and Diodorus give.
H. viii. 76. “The west wing put out and made a circling movement
towards Salamis.” It has been already pointed out that by “west
wing” Herodotus evidently means, not the west wing in the original
formation, but the west wing when the fleet had completed the
movement, and had taken up the position which he imagined it to
have assumed when the movement was complete. This “west wing”
would be the east wing in the original position. That it cannot have
been the original west wing has been pointed out in a previous note.
If this correction be made, Herodotus’ language in describing this
movement is peculiarly applicable to the movement of that part of the
Persian fleet which entered the strait by the channel east of
 Psyttaleia—ἀνῆγον κυκλούμενοι πρὸς τὴν Σαλαμῖνα; and the
applicability becomes still more striking in view of the evidence,
which will be given later, that this wing of the Persian fleet got in
advance of the other.
The left wing, which would use the channel west of Psyttaleia, is
equally referred to in the words: “Those about Keos and Kynosura
put out in order,” to which he adds, in accordance with his knowledge
that part of the object of the night-movement was the blocking of the
straits, “And they occupied the whole strait as far as Munychia with
their ships.”
8
This phenomenon of the morning wind is very common in the Greek
seas. It will be remembered that Phormio based his tactics in his first
battle with the Corinthian fleet just outside the Corinthian gulf on its
occurrence. I have experienced it there; and on the three occasions
on which I have been through the Strait of Salamis, once in the
summer of 1895, and twice in the summer of 1899, I have
experienced it on each occasion. It began in all three cases quite
suddenly, a little before seven in the morning, blowing from the west,
right down that part of the strait south of Ægaleos. It was extremely
violent while it lasted, though it did not raise a dangerous sea. To the
inexperienced it gave the impression that it meant the beginning of a
very windy day. On two occasions it ceased about 8.30, on the other,
shortly after nine, and the dead calm by which it had been preceded
ensued once more.
9
As is shown by the presence of an Attic vessel opposite the Persian
left, where her ships must almost certainly have been.
0
Cf. Æsch. Pers. 724,—Ναυτικὸς στρατὸς κακωθεὶς πεζὸν ὤλεσε
στρατόν. Thuc. i. 73, 5.—Νικωθεις γὰρ ταῖς ναυσίν ὡς οὐκέτι αὐτῷ
ὁμοίας οὔσης της δυνάμεως κατὰ τάχος τῷ πλέονι τοῦ στρατοῦ
ἀπεχώρησεν.
Modern historians have taken this account of the intended or
attempted construction of the mole too seriously. It has been pointed
out, for instance, that the only point in the strait east of the bay of
Eleusis at which it could possibly be carried out, is at the narrows
 where the island of St. George contracts the width of the channel,
and that it is impossible that, under the circumstances as they stood,
Xerxes should have been able to bring vessels to that part of the
strait. But Herodotus never attempts to give the impression that the
operation was ever undertaken seriously; he makes it plain, indeed,
that it was not. If that were so, and it was merely designed to give
the Greeks a wrong impression, it did not in the least matter whether
it was made at a possible or impossible point. Ktesias, Pers. 26, and
Strabo, 395, say that the mole was begun before the battle. This
would imply that a serious attempt was made to construct it. The
notorious unreliability of Ktesias, and the lateness of Strabo’s
evidence, render this account of the matter unworthy of
consideration.
2
H. viii. 103. Λέγουσα γὰρ ἐπετύγχανε τὰ πὲρ᾿ αὐτὸς ἐνόεε.
3
Οὐδεμία συμφορὴ μεγάλη ἔσται σεό τε περιεόντος καὶ ἐκείνων τῶν
πρηγμάτων περὶ οἶκον τὸν σόν.
4
It has been suggested that the real intention was to induce the
Ionians to revolt. The behaviour of this contingent in the recent battle
was not calculated to encourage such a plan, conceived within a few
days of the actual fight.
5
Ἐπείτε οὐκ ἐπαύετο λέγων ταῦτα ὁ Τιμόδημος, etc.
6
May it not be suggested that some archæologist acquainted with the
extant remains of Phœnician Carthage might confer a distinct
service on history by examining the structures at Agrigentum which
date from this period? The workman as well as the designer must
have set his mark there.
7
It has already been remarked that his description of Thermopylæ is
that of a traveller coming from the north—“from Achaia”—as he
himself says.
8
Herodotus himself (ix. 8) takes this view of the matter. He implies
that the Spartans did not care whether the Athenians medized or not
after the wall was completed. It is quite out of the question, however,
 to suppose that the Spartans could have regarded with equanimity
the possible transference of the Athenian fleet to the Persian side.
They had the experience of Artemisium and Salamis to guide them.
9
It is sometimes assumed from H. vii. 229, that the usual quota was
one helot to each hoplite; but a more probable interpretation of that
passage is that the reference is to the personal armed servant who
accompanied each hoplite to war, and that it cannot be deduced
therefrom that the body of these formed the whole number of the
helots present on an ordinary occasion.
Modern criticism of the impossibility of despatching so large a
force unknown to the Athenian embassy is not convincing. We do
not know the place at which it gathered. It is extremely likely that a
large number of helots were drawn from Messenia, and joined the
army at Orestheion, where the great route from Messenia meets the
route from Sparta by way of the valley of the Eurotas.
0
His departure from the Isthmus is ascribed by Herodotus to the fact
that when he was sacrificing ἐπὶ τῷ Πέρσῃ an eclipse of the sun took
place. This eclipse has been calculated to have occurred on the 2nd
of October, 480. If so, it would be about the time of the Persian
retreat from Attica after Salamis, and Stein’s conjecture that the
sacrifice had something to do with a plan to harass the Persian
retreat, has a certain amount of probability in its favour.
If Sparta had been careless as to whether Athens medized or not,
she might, probably would, have despatched troops to the Isthmus at
an earlier date. But if she was waiting until pressure of
circumstances forced Athens to adopt Peloponnesian views as to the
line of defence, then the delay is accounted for. Had her army been
at the Isthmus when Mardonius advanced into Bœotia, the
Athenians would certainly have called upon it to carry out the
agreement, and march to the northward of Kithæron. In that case the
Spartan government would have been obliged either to comply, or,
by a refusal, to show in the most unmistakeable manner possible the
war policy which it intended to adopt.
2
I was, I confess, surprised to find in August, 1899, that, in spite of
the excellent road to Megara from Bœotia by the way of Eleusis, the
track on the old line of the Platæa-Megara road is still largely used.
3
A road has been constructed through it in recent years, running from
Kriekouki on the Bœotian side to Villa on the south of the range.
4
I am inclined to think that the site of Skolos is that which Leake, and
others following him, have identified with Erythræ. Paus. ix. 4, 3,
says that if before crossing the Asopos river on the road from Platæa
to Thebes, you turned off down the stream, and went about forty
stades, i.e. four and three-quarter miles, you came to the ruins of
Skolos. This would place it not far east of the road from Thebes to
Dryoskephalæ. He speaks of Skolos in another passage as a village
of Parasopia beneath Kithæron, a rugged place, and δυσοικητός.
That seems to preclude the idea of its being near the river, which
traverses alluvial lands at this part of its course. The ruins identified
by Leake as Erythræ cannot belong to that town if the testimony of
Herodotus and Pausanias is accurately worded. This point will be
discussed in a later note. In actual fact, however, the exact site of
Skolos is very difficult to determine. My main reason for suggesting
that it stood where Leake places Erythræ is that those ruins are the
only ruins in the neighbourhood indicated by Pausanias, and are
certainly not the ruins of Erythræ.
5
It is necessary to pursue so obvious a line of argument, because, for
some incomprehensible reason, modern historians have thought it
right to judge of the plans of these able Persian commanders as
though they were dictated by no higher considerations than such as
might occur to an untutored savage.
6
The weakness of this line in case of attack from the north was
conclusively shown twenty years later in the manœuvres which led
to the battle of Tanagra.
7
It is almost certain that an ancient road from Eleusis followed the
eminently natural line taken by the modern road from Eleusis to
Eleutheræ. There was also, in all probability, a route from Athens to
 Eleutheræ which did not enter Eleusis at all, but, branching from the
Sacred Way near the Rheitoi after traversing the low pass through
Mount Ægaleos, went up the Thriasian plain and joined the road
from Eleusis among the low hills of Western Attica.
8
These ridges will be found numbered 1, 2, 3, 4, 5, 6, in the
accompanying map.
9
Marked A 6 in the map.
0
Called in the map, for purposes of distinction, the Asopos ridge, the
Long ridge, and the Plateau.
Those of the streams marked A 4 and A 5 on the map.
2
During my stay at Kriekouki, in December ’92–January ’93, the
rainfall was at times extraordinarily heavy. Nevertheless, I had not on
any occasion the slightest difficulty in crossing any of the streams,
and it was not even necessary to get wet in so doing. On one
occasion also I happened to be following the line of one of the
watercourses leading to the Œroë amid a downpour of rain such as
we rarely see in England, which had been going on with more or less
continuity for the previous fourteen hours; and yet, as I descended
the brook towards the plain the water became less and less until, on
the plain, there was no water running in the stream bed.
3
Pausanias knew the roads through these two passes.
(1) Platæa-Athens road.
He says (xi. 1, 6) that Neokles, the Bœotarch, in his surprise of
Platæa in the year 374, led the Thebans οὐ τὴν εὐθεῖαν ἀπὸ τῶν
Θηβῶν τὴν πεδιάδα, τὴν δὲ ἐπὶ Ὑσιὰς ἦγε πρὸς Ἐλευθερῶν τε καὶ
τῆς Ἀττικῆς.
There will be occasion to show that Hysiæ was in all probability a
small place, on a site just outside the southernmost end of the village
of Kriekouki. It was therefore at the eastern side of the opening of
the valley through which the road from Platæa to Athens passed.
The remains of that ancient road are, however, at the other side of
 the valley opening; and, therefore, Hysiæ was not upon it. Probably,
however, down the valley came a track which is still used, and which,
after passing through the village of Kriekouki, goes due north to
Thebes in a line parallel to the main road from Dryoskephalæ. This
would be the road which Pausanias here mention. It would, in
entering the valley to the pass, go close to this site of Hysiæ. Of the
identity of this site it will be necessary to speak in a later note.
In 379, after the revolution in Thebes (X. H. v. 4, 14), the Spartans
despatched Kleombrotos with a force to Bœotia. As Chabrias, with
Athenian peltasts, was guarding “the road through Eleutheræ,” he
went, κατὰ τὴν εἰς Πλαταιὰς φέρουσαν.
This is almost certainly the Platæa-Athens pass. Kleombrotos
probably did not discover that the Dryoskephalæ pass was guarded
until he got to Eleutheræ. After doing so he turned to the left and
made his way through the Platæa-Athens pass, exterminating a
small body of troops which attempted to defend it.
(2) The Platæa-Megara road.
Pausanias (ix. 2, 3) says, Τοῖς δὲ ἐκ Μεγάρων ἰοῦσι πηγή τέ ἐστιν
ἐν δεξιᾷ καὶ προελθοῦσιν ὀλίγον πέτρα· καλοῦσι δὲ τὴν μὲν
Ἀκταίωνος κοίτην.
In the previous sentence he has expressly spoken of the road from
Eleutheræ to Platæa. The Megara road is therefore a different road.
The κοὶτη Ἀκταίωνος can, I think, be determined with sufficient
certainty at the present day. It is on the top of a low cliff, probably the
πέτρα mentioned, overhanging the sources of the stream O 3. Near
the foot of the cliff is an ancient well, known in Leake’s time as the
Vergutiani Spring.
4
Ἐπὶ τῆς ὑπωρέης τοῦ Κιθαιρῶνος.
5
The site of Erythræ.
Colonel Leake identified it with certain ruins which are found at the
foot of the mountain slope several miles east of the road from
Dryoskephalæ to Thebes. The available evidence seems to me to be
strongly against this view.
 (1) The traditional site is where I have placed it, though I am afraid
that but little stress can be laid on traditions in modern Greece.
(2) Its comparatively frequent mention by Greek writers seems to
indicate that, though a small place, its position was of some
importance. If Leake’s view be correct this cannot have been the
case. If it were where I believe it to have been, it would be at the
northern exit of one of the most important passes in Greece. There is
an ancient φρουρίον on the bastion of Kithæron to the east of the
site. Its remains are so scanty, however, that they do not afford any
clue as to its date.
(3) There are remains of ancient buildings on the site. There are
also remains of an ancient well, besides which is a heap of stones,
from which two stones were obtained a few years ago with
inscriptions showing them to have belonged to a temple of
Eleusinian Demeter. Pausanias mentions so many temples in the
neighbourhood dedicated to that deity, that the discovery contributes
but little to the identification of the site. I was informed at Kriekouki
last year (August, 1899) that those particular stones were known to
have been originally discovered on another site. As neither my
informant nor any one else could tell me whence, why, or by whom
they were removed, I did not place much credence in the report.
(4) Pausanias says (ix. 2, 1), Γῆς δὲ τῆς Πλαταιίδος ἐν τῶ
Κιθαιρῶνι ὀλίγον τῆς εὐθείας ἐκτραπεῖσιν ἐς δεξιὰν Ὑσιῶν καὶ
Ἐρυθρῶν ἐρείπιά ἐστι; and further on (ix. 2, 2), he says, referring to
the road of which he is speaking: αὕτη μὲν (i.e. ὅδος) ἀπ’
Ἐλευθερῶν ἐς Πλάταιαν ἄγει. The road referred to is of course the
Athens-Platæa road, on which he is travelling towards Platæa. Can
any one suppose that Pausanias would have used the expression
quoted, especially the word ὀλίγον, had the ruins of Erythræ, as
Leake conjectured, lain some three and a half miles away from the
nearest point of this road, and hidden from it, moreover, by the great
projecting bastion of Kithæron, which is shown at the south-east
corner of the accompanying map?
Leake quotes Thucydides (iii. 24), who says that the two hundred
and twelve fugitives from Platæa first took the Thebes road in order
to put their pursuers off the scent, and then turning, ᾔεσαν τὴν πρὸς
τὸ ὄρος φέρουσαν ὁδόν ἐς Ἐρύθρας καὶ Ὑσιάς, καὶ λαβόμενοι τῶν
ὀρῶν διαφεύγουσιν ἐς τὰς Ἀθήνας. Meanwhile the pursuers were
searching the road along the ὐπωρέη. This last road would lead the
pursuers near the site where I conjecture Hysiæ to have stood, and
the objection may be raised that it is unlikely that the fugitives would
have gone to a place close to the road along which they could see
the pursuers were searching for them. It is, however, to be remarked
that Thucydides does not say that they went to either Erythræ or
Hysiæ. Had he intended to imply this he would have mentioned
those places in their proper order, Hysiæ first and Erythræ second.
Whenever he refers to the actual course taken by a body of men, or
by a fleet, he invariably mentions the places touched at or arrived at
in their geographical order. Vide Th. ii. 48, 1; ii. 56, 5; ii. 69, 1; iv. 5,
2; vii. 2, 2; vii. 31, 2.
The passage seems perfectly comprehensible and in accord with
the hypothesis which I put forward with respect to the positions of
Hysiæ and Erythræ. These fugitives, turning from the Platæa-
Thebes road, took the track which in modern times leads from
Pyrgos to Kriekouki, and which in ancient times would be the road
from Thespiæ to Hysiæ, Erythræ, and the passes. They did not go to
but towards those places, making in reality for those high rugged
bastions to the north-east of the pass of Dryoskephalæ.
But, after all, Pausanias’ words in the passage quoted dispose
effectively of Colonel Leake’s site. He would not have described a
place twenty-five stades away from the road as a short distance to
the right of it.
(5) Herodotus (ix. 15) speaks of the Persian camp as ἀρξάμενον
ἀπὸ Ἐρυθρέων παρὰ Ὑσιάς, κατέτεινε δὲ ἐς τὴν Πλαταίιδα γῆν.
These words merely show that Erythræ was east of Hysiæ.
(6) Perhaps one of the strongest pieces of evidence is Herodotus’
statement that the first Greek position was “at Erythræ.” Is it
conceivable that the Greek force, especially in its then state of
feeling with regard to the Persians, would be likely, after issuing from
the pass of Dryoskephalæ, to turn east along Kithæron, leave the
 pass open, and take up a position with their backs to a part of the
range through which there was no passage of retreat?
(7) We are told later that their reason for moving to their second
position was the question of water-supply. This accords with the
present state of the locality about the traditional Erythræ. The
streams in that neighbourhood have but little water in them in the dry
season.
(8) The ground in this neighbourhood accords peculiarly with the
description given by Herodotus of the first engagement.
6
Marked ridges 1, 2, 3, 4, in the map.
7
These positions will be found marked upon the accompanying map.
It is necessary, however, to explain the evidence on which they are
determined.
8
The details of the contingents given by Herodotus are:⁠—
Lacedæmonians—
Spartans 5000
Periœki 5000
Helots 35,000
Tegeans 1500
Corinthians 5000
Potidæans 300
Orchomenians (Arcadia) 600
Sikyonians 3000
Epidaurians 800
Trœzenians 1000
Lepreans 200
Mykenæans and Tirynthians 400
Phliasians 1000
 Hermionians 300
Eretrians and Styreans 600
Chalkidians 400
Ambrakiots 500
Leukadians and Anaktorians 800
Paleans from Kephallenia 200
Æginetans 500
Megareans 3000
Platæans 600
Athenians 8000
Miscellaneous light-armed troops 34,500
Total 108,200
9
I.e. A 1. In the days before scientific survey there was frequently the
utmost confusion with regard to the application of names to the head
streams of main rivers. This generally took the form of applying the
name of the main stream to several of its feeders. The tendency of
the local population was to apply the well-known name to that upper
tributary which was in their immediate neighbourhood, and was
therefore best known to them. Examples of this are frequent in
England; the upper waters of the Thames are a case in point. In
early sketch maps it will be found that the name Thames is applied
with the utmost diversity to the head streams of the river, and even a
tributary so far down as the Evenlode is sometimes given the name
of the main river. This is, I fancy, what has taken place with regard to
the Asopos. The Platæans, with whom Herodotus must have come
in contact in the course of his visit to the region, called this stream, A
1, by the name of the main river, and consequently “Asopos” in
Herodotus is to be understood to mean this stream up to its junction
with the stream which comes from the west, rising not far from
Leuktra, and, after that, to refer to what is really the main river. From
Platæa itself the course of this stream is plainly traceable in the
 plain, running along the western base of the Asopos ridge. The
stream coming from Leuktra is not visible, and it is quite conceivable
that Herodotus never had any definite knowledge of its existence. In
Leake’s time (vide his sketch map) the inhabitants of Kriekouki seem
to have called the stream, A 6, Asopos. It is not so called at the
present day. My own impression is, however, that Herodotus,
although he heard the Platæans speak of A 1 as the Asopos, may in
one passage refer to the stream from Leuktra with a special attribute:
τὸν Ἀσωπὸν τὸν ταύτῃ ῥέοντα (H. ix. 31). A sentence previously, at
the end of Chapter 30, he has a reference to the Asopos without any
qualification, οὗτοι μὲν νὺν ταχθέντες ἐπὶ τῷ Ἀσωπῷ
ἐστρατοπεδἐυοντο, and this reference is undoubtedly to A 1, which is
to him, as other references in his narrative show, the upper Asopos
“ordinarily so called.”
0
H. ix. 31, ad init., πυθόμενοι τοὺς Ἕλληνας εἶναι ἐν Πλαταιῇσι.
Cf. especially the mention of the Asopos and its context in Chapter
40.
2
It will be remarked that Artabazos’ statement on this point is in direct
conflict with that reported by Herodotus to have been made at the
same time by Alexander of Macedon to the Greeks.
3
It appears later (Chap. 46, ad init.) that it was to the Athenian
generals alone that Alexander’s story was in the first instance
imparted. That tends to confirm, what the lie of the ground would
suggest, that the Greek left was nearer the Asopos than the right
wing.
4
This is one of the most important passages in Herodotus’ description
of the battle. It indicates more clearly than has been hitherto
indicated, the position of the Greeks in their second position.
In the first place, if we remember that the Lacedæmonians were
on the Greek right, it will be seen that it forms a very strong
argument in favour of the identification of Gargaphia which has been
adopted. Had it been at Apotripi it would certainly have been near
the Greek centre. It also shows the obliquity of the Greek line with
 respect to the course of the Asopos; in other words, that it was, as
might be expected, extended along the Asopos ridge.
5
This is shown still more clearly in the account of the withdrawal from
this position.
6
The three developments of the Greek second position may be
summed up as follows:⁠—
1. The Greek right was near the spring of Gargaphia, not on the
Asopos ridge, while the left was near the Heroön of Androkrates.
2. After a forward movement of the whole line, the right took up
position on the Asopos ridge, while the line extended along the
course of that ridge, until the left was actually on the Asopos.
3. The left, when its position on the plain became untenable, took
to the higher ground of the north extension of the Asopos ridge.
7
It would seem as if this determination were not come to at the
morning council. Their idea at that time appears to have been to
move during the night, in case the enemy did not renew their attack.
As the attack was renewed, the movement was deferred until the
following night.
8
The members of the American school at Athens who excavated
parts of the site of Platæa some years ago were inclined to believe
that at the time of the battle the town stood on the higher or southern
end of the bastion which is now strewn with the traces of the
successive towns which have occupied the site; and that it did not
extend northward to the point where the bastion sinks more or less
abruptly into the plain. They also believed that they discovered the
foundations of the temple of Hera on this north extension of the
bastion. I am disposed to think that their conjecture as to the position
of the contemporary town is correct, though the question is not of
sufficient importance with respect to this particular passage in Greek
history to render it desirable or necessary to quote the mass of
evidence on which the opinion is founded. The position of the temple
of Hera as determined by them agrees with the brief mention of it in
this passage of Herodotus.
9
Herodotus, in words already quoted, says that it was the intention of
the Greeks, on moving to the “Island,” to detach a part of the army to
relieve the attendants who were blocked in the pass. This is certainly
the Dryoskephalæ or the Platæa-Athens pass, probably the latter,
which they were attempting to use as an alternative way, after the
fearful disaster which had befallen the former provision train in the
exit of the Dryoskephalæ pass. Herodotus shows, too, that this relief
was urgently required, since the Greek army was running short of
provisions; for, although the Platæa-Megara pass must have been
open, it is of such a character as to render it impossible that the
commissariat for a force of 100,000 men could be adequately
maintained through its channel. It is therefore in the very highest
degree probable that an attempt, at any rate, was made to carry out
this part of the arrangement between the generals. Now, the Spartan
force on the right of the Greek line would be, in so far as position
was concerned, that portion of the Greek army on which this duty
would naturally devolve. The mission of this force for the relief of the
pass was one of extreme danger and difficulty, and it would be
natural that the service should devolve on that part of the army which
enjoyed the highest military reputation. It was, I venture to think,
while carrying out this movement that the Spartans became involved
in that series of events which led to the last catastrophe in the great
tragedy.
0
Thucydides (i. 20) denies that such a division or regiment existed in
the Spartan army.
Even in the Spartan army indiscipline was apt to make its
appearance without the existence of such a substantial motive as in
the present instance. Cf. the insubordination of the Spartan officers
at the battle of Mantinea in 418 b.c. (Thuc. v. 72).
2
That they never reached the rocky ὑπωρέη is plain from the
incidents of the battle that followed.
3
Of A 4 and A 5.
4
The ὑπωρέη of Herodotus.
5
Cf. the tale H. ix. 58.
6
Δρόμῳ διαβάντας τὸν Ἀσωπὸν (H. ix. 59).
7
It will be seen, when the details of the Athenian retreat come to be
examined, how noticeably this detail accords with the account which
Herodotus gives of that retreat.
8
Some modern commentators have regarded this detail mentioned by
Herodotus as a convincing proof of the Athenian bias in his narrative.
To me it seems eminently natural, after the experience of the
previous days, that Pausanias or any other commander should have
summoned help under the circumstances. I shall, moreover, have
occasion to show that the Athenians did undoubtedly diverge from
their march to the Island in the direction in which the Spartan battle
with the Persians took place.
9
It is clear from Herodotus’ subsequent account of the proceedings of
the Greek centre that this battle took place out of sight of that part of
the army which had retired to Platæa.
0
H. ix. 62: Ἤδη ἐγίνετο ἡ μάχη ἰσχυρὴ παρ’ αὐτὸ τὸ Δημήτριον.
This incidental detail mentioned by Herodotus peculiarly supports
the view that the temple must have stood on the site of the church of
St. Demetrion.
2
This is clearly shown in Herodotus’ narrative. He distinctly speaks of
the Athenians as having at the beginning of the movement “turned
down towards the plain” (H. ix. 56, κάτω τραφθέντες ἐς τὸ πεδίον);
and in a still more remarkable passage he says that, when
Mardonius led his Persians across the Asopos in pursuit of the
Greeks, “he did not see the Athenians, who had turned down
towards the plain, by reason of the (intervening) hills” (H. ix. 59). The
hills mentioned are evidently the northern extension of the Asopos
ridge.
3
A 1 in the map.
4
Ridge 5.
5
Thus far διὰ τῆς ὑπωρέης (H. ix. 69).
6
I.e. ridges 3 and 2; cf. H. ix. 69, διὰ ... τῶν κολωνῶν.
7
Ridge 5.
8
I confess I cannot understand the argument of those who regard
Herodotus’ account of Platæa as being tainted throughout with a
lying Athenian tradition. In so far as the narrative provides evidence
of its source or sources, there is at least as much matter in it which
may be attributed to Spartan as to Athenian origin.
9
The Asopos ridge, the Long ridge, and the Plateau.
0
The treatment meted out to the Æginetans in the narrative of
Platæa, as contrasted with the account which Herodotus gives of
their conduct at Salamis, points to the very various character of the
sources from which he drew his history. This part of the Platæan
narrative is undoubtedly drawn from a tradition highly coloured by
the relations which existed between Athens and Ægina twenty years
after Platæa was fought.
Xen. Anab. iii. 2, 27. The striking words are μὴ τὰ ζεύγη ἡμῶν
στρατηγῇ.
2
I have had occasion to speak of the Thermopylæ narrative under
various aspects in relation to the sources from which it is derived.
To prevent any misconception, I should like to sum up briefly my
conclusions.
(1) The whole “motivation” of the story is derived from a
version of official origin at Sparta.
(2) The incidents of the actual fighting may be derived partly
from a Spartan source, probably of an unofficial character. The
description of some of them, however, rests on information
picked up by Herodotus at Thermopylæ itself from natives of the
region.