Buffon's needle and Buffon's noodle

The probability distribution of the number of crossings depends on the shape of the noodle, but the expected number of crossings does not; it depends only on the length L of the noodle and the distance D between the parallel lines (observe that a curved noodle may cross a single line multiple times).

This fact may be proved as follows (see Klain and Rota). First suppose the noodle is piecewise linear, i.e. consists of nstraight pieces. Let X_i be the number of times the ith piece crosses one of the parallel lines. These random variables are not independent, but the expectations are still additive due to the linearity of expectation:

${\displaystyle E(X_{1}+\cdots +X_{n})=E(X_{1})+\cdots +E(X_{n}).}$  (wiki)