In my December diary I posed the following brain-teaser.
I have a circular disk, radius one unit. I pick two points at random on the disk and measure the distance d between them. I repeat this process some large number of times. What's the average value of d?
Here we are in the tiny but fascinating realm of Geometric Probability, which was launched when Georges Buffon posed his famous Needle Problem in 1733. Subsequent key names in the topic are Morgan Crofton (19th century) and Luis Santaló (20th). The textbook I own is Herbert Solomon's 1978 production; but people tell me there are better ones now. The main drawback of Solomon's book is that it has no exercises. It explains things pretty well, though.
Geometric probability is also well covered on the internet. In fact Wolfram's MathWorld has the solution to this puzzle. Answer: 128/(45π), which is 0.905414787367226799 …
• 1: For random points inside the disk, coordinates x = r cos θ, y = r sin θ won't do. You need the square roots of random r's in the coordinates. This is explained here.
• 2: Having picked your first random point, you can wolog rotate the disk so that this first point is on the positive x-axis at (√r1, 0).
• 3: Writing the second point as (√r2, θ), the distance d between the two points is given by the cosine rule:
d 2 = r1 + r2 − 2√r1r2 cos θ
• 4: Now you just have to add up all possible values of d and divide by their number. True, numerator and denominator are both uncountable infinities … but that's what integration's for! Hence Weisstein's formula (3) for the average on that MathWorld page, and the result follows.