Download 1 Kroesus and the oracles

1. Kroesus and the oracles
Kroesus and the oracles
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1.1 The historical background
king of the Lydians, was doing just then. Only the
oracle in Delphi and the oracle of Amphiaraos gave
correct answers and were thereby judged to be true
oracles.
To these two Kroisos sent new messengers in order to ask if he should march against the Persians.
Both oracles predicted that if Kroisos crossed the
river Halys a great realm would be destroyed 1 . This
Kroisos took as good sign and marched against the
Persians . . .
Later, Kroisos sent a complaint to the oracle in Delphi, accusing it of giving false predictions. The oracle answered that he only had himself to blame for
not asking another, well motivated, question.
(See, e.g. Herodotos Historia: First book, in particular
chapters 46–49, chapter 53 and chapter 91. An English
translation (with the Greek original) can be found here:
http://www.perseus.tufts.edu/Texts.html
under ”Ancient Greek Texts”, ”Herodotus. The Histories”.)
In the middle of the 6th century B.C. the Persian realm started to expand. Kroisos (Kroesus),
king of Lydia c:a 560–546 B.C., was worried by
the developement. Lydia then covered what is now
western Turkey, between the Hellespont and the
river Halys which reaches almost from the coast
opposite Cyprus up to the Black Sea. Therefor, he
wanted some advice regarding whether it would be
a good idea to attack Persia. The usual way to receive such advice was to apply to one of the many
oracles around the world.
Since Kroisos was aware that not all oracles were
true oracles he desided to first ask a question that
would be almost impossible for a false oracle to
guess the answer to. To that end, he sent out
messengers to six oracles in Hellas (Delphi, Abe in
Phokis, Dodona, Amphiaraos, Trophonios and the
Brankids in Milesia) as well as to the oracle of Ammon in Libya, instructing them to, on a particular
day, ask the oracle what Kroisos, son of Alyattes,
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1.2 Some assumptions
We assume that a true oracle always gave correct
answers and true predictions while the chances of a
false oracle giving a correct answer varied with the
difficulty of the question.
If the question was “Should Kroisos march against
the Persians?,” the false oracle would have a 50%
chans of a correct prediction by tossing a coin. The
coin was invented in Lydia 150 years previously,
a fact that might be a contributive reason for the
great wealth of Kroisos.
If there were several alternative answers the chances
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1. Kroesus and the oracles
of a correct guess were smaller. We assume the
chances of a correct guess regarding what Kroisos
did that particular day to be 1 %.
We also assume that the chance of a correct guess
was independent of whether the oracle made correct guesses at other questions and also that differ-
ent oracles were independent of each other.
We also need to know the precentage of true oracles. We assume that every fifth oracle was true and
regard the seven oracles Kroisos chose as a random
sample of all possible oracles.
1.3 Some questions
(i) What is the probability that an oracle was
true, given that it was correct at both occasions?
The filtering procedure
(a) What is the probability that a randomly chosen oracle could giva a correct answer to the
question of what Kroisos was doing that particular day?
(j) What is the probability that both the oracle
in Delphi and the oracle of Amphiaraos were
true, given that they were correct at both occasions?
(b) What is the probability that an oracle was
a true oracle given that it gave a correct answer?
The entire procedure
(c) What is the probability that both the oracle
in Delphi and the oracle of Amphiaraos were
true oracles given that they both gave correct
answers?
(k) What is the probability that all predictions
would be correct if Kroisos had asked seven
oracles if he should marsh against the Persians, without first filtering out obviously
false oracles?
The counselling procedure
(d) What is the probability that a false oracle
could both tell what Kroisos did that particular day and give a correct prediction of
what would happen if he marched against
the Persians?
(l) How large is the risk that Kroisos would not
have any oracle left after the filtering procedure?
(m) [Voluntary]
(e) What is the probability that a randomly chosen oracle could both tell what Kroisos did
that particular day and give a correct prediction of what would happen if he marched
against the Persians?
What is the probability that none of the false
oracles that might be persent would be able
to tell what Kroisos did that particular day?
(Condition on a suitable Binomial distribution and use Maple.)
(f ) What is the probability that a randomly chosen oracle could give a correct prediction,
given that it could tell what Kroisos did?
(n) [Voluntary]
What is the probability that Kroisos with
his filtering procedure would get any oracles
left, true or false, who whould then all give
correct predictions? (Condition on a suitable Binomial distribution and use Maple.)
(g) What is the probability that a randomly chosen oracle could give a correct prediction,
given that it could not tell what Kroisos did?
(h) What is the probability that a randomly chosen oracle could give a correct prediction, regardless of whether it could tell what Kroisos
did?
(o) [Voluntary]
What was it that Kroisos did that particular
day that was so difficult to guess?
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