[kittenfluff]

My favorite analysis of Buffon's needle is this one, which pulls out the correct probability of intersection without any calculus -

Consider a straight needle of length L dropped onto lines spaced 1 unit apart. It is clear that the expected number of intersections is proportional to L. If you joined two needles of length L/2 (possibly at an angle) the expected number of intersections is still L, by additivity of expectation (note that we don't need independence). By induction, the shape of the needle doesn't matter, and the expected number of intersections is proportional to L.

We can work out the constant of proportionality by considering a needle shaped like a circle of radius 0.5, which has circumference pi, and is guaranteed to intersect the lines in two places. Therefore

  2 = C * pi

and hence C = 2 / pi

Now for a straight needle of length 1, we can never have more than 1 intersection. Therefore the probability of intersection is the same as the expected number of intersections, which is seen to be

   P = (2 / pi) * 1 = 2 / pi