Sunday, 18 November 2012

A Farewell to the Bertrand Paradox


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.

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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?

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).

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?

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).

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?

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).

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 – if your selection is sufficiently random.  (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 do define a chord, you can throw away the failed pairs of points without causing any harm to your random distribution.

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:





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.

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

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 within 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 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.


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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 approaching a truly random distribution of chords.

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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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