Okay,
this will be the last one on the Bertrand Paradox.

I
really only intended it to be a bit of light-hearted stab at William Lane Craig
and his misuse of infinity. I didn’t
intend getting drawn into the nitty-gritty.
Anyways …

My
parting shot is an attempt to state my position. I apparently fall into the smaller of two schools
of thought on the paradox, that the probability in question is determined at p=0.5 rather than being fundamentally undetermined. If you don’t know what I am talking about
now, and are vaguely interested, you have to go back to where I started

*three posts ago*.
My
argument for p=0.5, developed after the work reflected in recent post, is not the standard argument – as far as I can
tell. So, I will detail what that
argument is here, as briefly as I can, then I’ll leave it at that.

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

Fundamentally
the argument is that if you restrict the options for selecting a random chord,
you are less likely to get a properly random distribution of chords and it is this that we see manifested as the “paradox”. The method for getting the best, most random
distribution (I believe) is to follow this procedure for determining a chord
for a given circle:

__My Method__

1. Select a random point.

2. Select a second random point.

3. Does the line defined by those two
points intersect a circle?

NO –
return to step 1.

YES
– continue.

4. You now have two points on the
circle, between which is a chord (or a vanishingly small chance of a tangent).

The
three classic methods of determining chords are as follows (using the order in
which they appear in the

*Wikipedia entry*):__Classic Method 1:__

1. Select a random point

**on the circumference on the circle**.
2. Select a second random point

**on the circumference on the circle**.
3. Does the line defined by those two
points intersect a circle?

YES
– continue.

4. You now have two points on the
circle, between which is a chord (or a vanishingly small chance of a tangent).

__Classic Method 2:__

1.
Select a random point

**on the circumference on the circle**.**1a. Draw a line segment between the random point and the locus of the circle.**

2.
Select a second random point

**on the radius just drawn**.**2a. Draw a line at the second random point, perpendicular to the radius.**

3. Does the second line drawn intersect
a circle?

YES
– continue.

4. You now have two points on the
circle, between which is a chord (or a vanishingly small chance of a tangent).

__Classic Method 3:__

1. Select a random point

**inside the circle**.**1a. Draw the line defined by that point and the locus of the circle.**

**1b. Draw the line perpendicular to the line just drawn at the random point.**

2. Select ~~a second random
point~~

**one of the points intersecting with the circle as the second point**.
3. Does the line defined by those two
points intersect a circle?

YES
– continue.

4. You now have two points on the
circle, between which is a chord ~~(or a vanishingly
small chance of a tangent)~~.

Now,
if you are an average person, this will mean little or nothing to you, but if
you are someone who grapples with philosophy of mathematics, or otherwise has
an interest in the Bertrand Paradox, you might notice that in all four methods (mine and the three classic methods), you end up with two points defining a chord in Step 4. But in the three classic methods, you have restrictions
on your selection of points and, in two methods, there are extra steps.

I am
fully cognisant of the fact that you won’t get a chord every single time with
my method, in fact almost all of the lines you draw won’t end up defining a
chord –

**. (In my simulation, with a limit on points as follows: (-100.r,-100.r)<=(x,y)<=(100.r,100.r), 99.5% of all pairs failed to define a chord.) But, since you are only interested in the lines that***if your selection is sufficiently random***define a chord, you can throw away the failed pairs of points without causing any harm to your random distribution.***do*
The
three classic methods, on the other hand, produce a chord every single time,
which is convenient. The cost of that
convenience however, in the case of methods 1 and 3, is a skewing of the
distribution. Method 2 has the benefit
of being convenient and also producing the result p=0.5, the result you arrive
at with the inconvenient method.

I should add, as a
clarification, that when Method 2 gives the correct result this appears to be a
consequence of what I think of as “maximal uniformity”. In other words, the result is not skewed because
the method lends itself to a smooth distribution of chords. Method 2 can be generalised to reflect
an aspect my method:

4. You now have two points on the
circle, between which is a chord (or a vanishingly small chance of a tangent).

__Method 2 (Generalised):__

1.
Select a random point.

**1a. Draw a line between the random point and the locus of the circle.**

2.
Select a second random point.

**2a. Draw a second line which is perpendicular to the first and also passes through the second random point.**

3. Does the second line intersect the
circle?

NO –
return to step 1.

YES
– continue.

Again,
this method will result in the vast majority of pairs of random points failing
to produce a chord. Chords will only be
produced if the second line intersects the first

**the circle. The classic Method 2, therefore, by restricting the selection in step 2 merely eliminates the vast majority of pairs of random points that would not result in a chord***within***.***without skewing the outcome*
The
classic Method 3 can have “maximal uniformity” imposed on it by using polar
co-ordinates rather than cartesian, as explained in

*an earlier article*, resulting in p=0.5.
--------------------------------

I am
aware that the Bertrand Paradox has value in that it demonstrates that poor
selection of a methodology can result in incorrect modelling of a random
distribution, and there may be other pedagogical or philosophical uses to which
it can be put. But I’m not convinced
that that value means that we should not accept that there is in fact a
uniquely correct, if somewhat inconvenient method of

**a truly random distribution of chords.***approaching*
--------------------------------

It
struck me a few days after I posted this that I could describe my argument differently,
as a sort of decision tree. Just to ram
the point home yet another way, here is the decision tree for my method (green),
method 1 (light orange), method 2 (blue) and method 3 (dark brown):

As
the count at Z approaches infinity, the values of E/D and F/D both approach
zero. (If you don’t believe me, get
yourself a random number generator and try it.)

The
restriction of hypothetical samples of chords per methods 1 and 3 is actually
quite severe. Imagining that such a
restriction won’t have any effect at all is hubris of quite staggering
proportions.

If
this doesn’t persuade detractors, then I guess nothing will.

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