If you take a set of random dots on a plane and then pick two more rnadom points on the paper, there's anice formula for calculating the likelyhood that the line between these two points is uninterupted by one of the original points. Obviously, by rejigging this you can calculate the likelyhood that the line is interupted by a point ( one minus the first value ).
The formula is actually a function of PI, which is nice and just goes to show that nature has a sense of humour :-)
I'm trying to dig the formula out of all my old maths books. From what I can remember the likelyhood that the line is uninterupted is quite low, which means that the likelyhood that it will be interupted is quite high. Thus making alignments of three points in a large set quite likely by coincidence.