When Buffon meets Bertrand

When Peter Diggle gave his “short history” of spatial statistics this morning (I typed this in the taxi from Charles de Gaulle airport, after waiting one hour for my bag!), he started with a nice slide about Buffon’s needle (and Buffon’s portrait), since Julian Besag was often prone to give this problem as a final exam to Durham students (one of whom is responsible for the candidate’s formula). This started me thinking about how this was open to a Bertrand’s paradox of its own. Indeed, randomness for the needle throw can be represented in many ways:

• needle centre uniformly distributed over the room (or the perpendicular to the boards) with a random orientation (with a provision to have the needle fit);
• needle endpoint uniformly distributed over the room (again a uniform over the perpendicular is enough) with a random orientation (again with a constraint);
• random orientation from one corner of the room and a uniform location of the centre on the resulting line (with constraints on both ends for the needle to fit);
• random orientation from one corner of the room and a uniform location of one endpoint on the resulting line, plus a Bernoulli generation to decide on the orientation (with constraints on both ends for the needle to fit);
• &tc.

I did not have time to implement those different generation mechanisms in R, but have little doubt they should lead to different probabilities of intersection between the needle and one of the board separations. I actually found a web-page at the University of Alabama Huntsville addressing this problem through exercises (plus 20,000 related entries! Including von Mises‘ Probability, Statistics and Truth itself. A book I should read one of those days, following Andrew.). Note that each version corresponds to a physical mechanism. Thus that there is no way to distinguish between them. Had I time, I would also like to consider the limiting case when the room gets infinite as, presumably, some of those proposals would end up being identical.