Interaction
between Melchior and me at

*reddit/math*has got too long to continue in its original form. Here’s what’s gone before in response to*my recent article about the Bertrand Paradox*(note that by some cruel twist of fate, the name neopolitan was not available to me):
---------------------

**Melchior:**

You
pose an ill-defined problem to which there is no right or wrong answer, then
you nominate one answer as "probably right" for dubious reasons. You
stick with the vague "select at random from a set" concept and try to
work with subset relationships, but this approach can't express the hidden
assumptions that underlie the paradox.

---------------------

**wotpolitan:**

Thanks
Melchoir.

I
posed the question as "what is the probability, p, of a random chord in a
circle being longer than the side of the largest equilateral triangle that can
be drawn in that same circle?" Wikipedia has it expressed as
"Consider an equilateral triangle inscribed in a circle. Suppose a chord
of the circle is chosen at random. What is the probability that the chord is
longer than a side of the triangle?"

I
don't really think that either question is ill-defined.

It
only appears ill-defined when you select chords from a biased set (thus making
them not random).

My
set was only biased so as to make sure the two random points result in a chord,
I don't think that that is invalid, is it?

But,
yes, I agree, I do have to make that a little more clear in the original article.
Thanks again.

---------------------

**Melchior:**

| “It
only appears ill-defined when you select chords from a biased set”

There's
no such thing as a "biased set". Every probability distribution is
"biased" relative to some other probability distribution.

| “(thus
making them not random)”

The
phrase "not random" is meaningless here. I think you mean that the
distribution is not uniform, but that's also meaningless unless coordinates are
specified, and it does depend on the coordinates.

---------------------

**wotpolitan:**

Ok,
maybe this is why I should have paid a little more attention to statistics many
years ago.

Re a
“biased set” – if you are considering the probability of selecting a human over
180cm tall and you restrict your measurements to all American males, then you
are not going to get the correct result for all humans. This is what I mean by
a “biased set”. You might want to call this a “biased sample”, but note that I
said

**ALL**American males. If you, for some reason have got it in your head that the set of all humans is identical to the set of all American males, then you have a biased set.
This
is similar to the case with the p=1/3 calculation of the probability of
randomly selecting a chord of length greater than r.√3 in a circle of
radius r. If you restrict your selection of points which define a line which
then results in a chord to any two on the circumference of the circle, you get
a “biased set”. The biased set results in p=1/3 while an unbiased set (which I
think I’ve defined) results in p=1/2. I’m sorry if my terminology is not
standard but if you can try to see past that it would be much appreciated.

Re “

**not**random”, hopefully this is clarified by the above but to make sure, by this I mean that a random selection from the set of all possible chords is not going to give you the same result as the random selection of a pair of points on a circle, even if those points can then be used to define chords. A “random” selection from a subset of all chords is not necessarily equivalent to a random selection from the set of all chords. (It’s like trying to select people at random from a crowd and subconsciously only selecting females that you think are particularly good-looking, it might be a random selection of good-looking females, but it’s not random in terms of the task at hand.) Again, I might not have used the correct terminology for which I apologise.
By
the way, are you a p=1/3 person? Or simply undecided?

---------------------

**Melchior:**

The
set of humans is simply different from the set of American males. Neither one
is biased.

The
distinction is a red herring anyway, because we're talking about one and only
one set: the set of chords in the circle. You keep saying "subset",
but the set of chords defined by choosing two endpoints is precisely the same
set as the set of all chords. There is no chord that exists in one set but not
the other.

Subsets
can't rescue you from this paradox. It's not just a matter of terminology;
there's something very different going on here. This is a really important
point, and you have to accept it in order to understand anything else about the
problem.

---------------------

Dear
Melchior.

This
last response is quite interesting.

*Wikipedia defines a paradox*broadly enough to include puzzling results that can be rectified by demonstrating that one or more of the premises themselves are not really true, are a play on words, are faulty and/or cannot all be true together, going far enough to state that paradox is often used interchangeably with

*contradiction*. Even dictionaries seem to be rather vague on the topic, for example the first entry at

*merriam-webster.com*isn’t even remotely close to what I personally would define as a paradox.

Let me clarify. In this context I mean by paradox, as I am
sure the people who originally named it the

*Bertrand Paradox*meant, a problem which appears to have (at least) three alternative solutions which cannot all be correct. It cannot be true that p=1/2 and p=1/3 and p=1/4 in this scenario, when p is the probability that a chord selected at random is greater than r.√3.
I

**that Bertrand originally presented the problem as an example of how different methods of producing a random variable can affect the result, but I think you might be interpreting this incorrectly. A truly random selection of a chord is not necessarily equivalent to following a process which is (in a sense) random and which results in a chord. You might***know***that, by following each of the three methods listed in the Wikipedia article, you are randomly selecting a chord, but my argument is that you are not. You are close to it with method 2 (the radius method), but you are not with methods 1 or 3 – unless you modify them in the way that I have described in my earlier article.***think*
I am curious as to your intent,
Melchior. It may be a tiny bit unfair to
use this medium to analyse your responses, but life is always a tiny bit
unfair, so I’ll carry on regardless.

You
initially said the problem is ill-defined.
It’s not. You also made mention
of the “hidden assumptions that underlie the paradox”, but you’ve not made
clear what these hidden assumptions are.
Accepted that I haven’t previously asked you to, but I am now: What are the hidden assumptions underlying
the Bertrand Paradox?

Then
you totally misunderstood the point about the “biased set”. The set of American males is a set and the
set of humans is a set, it’s generally accepted that the former is a subset of
the latter. The problem arises when you
take only American males and assume that the results of any measurement will be
representative of humanity as a whole.
The distribution of chords as generated via two random points on the
circumference of a circle is not the same as the distribution of chords as
generated by two random points selected without the constraint. The former is a subset of the latter. Do you not agree?

I
did ask you if you were a p=1/3 person or just undecided, which you didn’t
respond to. Could you please be open
about that?

Finally,
you say that subsets “can’t rescue (me) from the paradox” and “there’s
something very different going on here”.
Then you talk about understanding “anything else about the problem”.

When
you talk about rescue from a paradox, it seems to me that you are talking about
inherent self-contradiction as in the following couplet:

The sentence below is
a lie.

The sentence above is
true.

That’s
an inescapable paradox caused by indirect self-reference, you have to just walk
away from it rather than try to work out whether one or both or neither of them
are correct.

Are
you implying that the Bertrand Paradox is a true paradox in the stricter
sense? Is the “something very different”
that you are hinting at merely a reference to the problems associated with
simulating randomness (which Bertrand was in essence talking about)?

And,
as a last question, what is the “anything else about the problem” of which you
speak?

Sincerely
in anticipation of a fruitful continuation of this discussion,

neopolitan

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