[enricozb]

Pretty neat! However, if you wanted to know the _probability_ of a noodle crossing any line in the long noodle case (L/W > 1), the expression is more complex (and I believe would require an integral) :).

It's interesting that the number of crossings is independent of whether L/W is less than or greater than 1, but the probability of crossings is equal to 2pi * L/W only in the short case. This makes sense since in the short case the noodle can at most cross a single line.

[_alternator_]

This is the crux of the observation. For needles of length less than W, the probability that it crosses a floorboard is equal to the average number of floorboards it crosses. (Exercise for the reader ;))

The point is that the "right" quantity to be considering for the problem is the average number crossings, since that naturally extends to curved noodles, lines of any length, and even circles. The number of crossings is also known as the Euler characteristic of the intersection, and there's a rather deep and beautiful theory of geometric probability that takes this as the jumping off point.

[gowld]

That's the menace of expected-value problems. They are easy-to-solve variants of more interesting hard-to-solve problems.

[_alternator_]

Is the probability actually more interesting though? I find the symmetry of this type of result extremely compelling, beautiful even. Buffon himself restricted his attention to the case where the needle was short enough that "probability" and "expectation" had the same answer. Put simply, math is best when complicated-seeming things suddenly become simple.