Three New Wrong Answers for Bertrand

I’ve already said a fond farewell to the Bertrand Paradox but, as is my wont, I got to thinking again.  What about if you split the circle in two, picked a point at random in one half, then picked a point at random in the other half, and drew the chord defined by those two lines.  What would be the likelihood that the chord created would be longer than √3r, where r is the radius of the circle?  This is, of course, equivalent to picking a chord at random, but it’s not one of the methods identified by Bertrand.  Even so, I would have expected the answer to be either 1/2, 1/3 or 1/4.

But it’s not, or rather not necessarily.  If we permit only points which fall in the circle – the equivalent of splitting the circle in half and picking random points in each half, the answer comes to 3/4.  If we permit points that fall in the boxes, the answer comes to 1/2 – the answer that I think is correct.

The odd thing is that this gives us another “wrong” answer, which I’ve not seen before.  But it gets worse.  Much worse.

If we don’t split the box in two, and simply select two random points in the unsplit box, and then draw the chord defined by those two points, we get two more “wrong” answers.

And they are very strange numbers.  Firstly, we have to eliminate all pairs of points that don’t define a chord (which only happens when both of them fall into the corners outside the circle).  This accounts for in the order of 0.5% of the pairs (closer to 0.06%).  We could go further and eliminate all pairs for which at least one does not fall in the circle.  If we do so, then about 44% of chords are longer than √3r.  If we don’t take that extra step, then about 65% of chords are longer than √3r (not 66.666%).

I’ve been wracking my brains trying to work out what these figures mean.  The 44% is close to 4/9 but even after 100,000 iterations, it’s still out by about 1% and the 65% is very close to 2/3 but still out by about 2% after 100,000 iterations.  I’m willing to accept that I’ve made an error somehow, but I can’t currently see it.  Even so, it looks like we’ve got two new “wrong” answers.

Comments?

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I lay out a bit of the maths here.