On a circumference of a given circle with a radius R two random points A & B are independently chosen.

What is the probability of AB being less than R ?

I find the idea of probabilities involving infinities quite disconcerting, and computer modelling is a bit beyond me these days.

There is an alternative involving a large but finite number of points, say n points, equally distributed around the circle.

http://perplexus.info/show.php?pid=11330&cid=59638 then provides a clue, because A can be considered as fixed under rotation; B can be any of the other points; and R may as well be 1.

Then the sines of the various lines are sin(90/(n-1)) to 90/(n-1) to 90(n-2)/(n-1), plus the diameter of value 2, and those are to be randomly chosen between.

If n is large, and the number of trials is also large, then the probability of AB being less than R approaches 1/3, for the simple reason that the length R corresponds to sin (30) = 1/2, namely half the diameter, 2, or 1.

So, say n=100, then chord 33 = sin(33*90/(100-1)) degrees = 1/2, so 32 of the possible chords will satisfy the condition; if n=1000, then chord 333 = sin(333*90/(1000-1)) degrees= 1/2, so so 332 of the possible chords will satisfy the condition, and so on.

Still no alternative value, unfortunately.

Edited on April 27, 2018, 7:32 am

Posted by broll on 2018-04-27 07:10:30