The following is an elementary probability question related to a generalization of the famous "Buffon's needle experiment" which allows one to estimate $\pi$ by counting how many times a randomly tossed needle crosses a line on a lined sheet of paper. If we replace the needle with a rigid wire in the shape of any piecewise smooth plane curve, I believe it is well-known that the expected number of line crossings depends only on the length of the wire and not on its specific shape.
I am seeking an elementary proof of this fact in the case where the wire consists of two line segments joined end-to-end. The only parameters here are the lengths of the two line segments and the angle at which they are joined; I would like to prove that the expected number of crossings depends only on the sum of the lengths. If it helps, I am happy to assume that both line segments are very small compared to the spacing between the lines on the paper. Any ideas?
Added: Several have argued that this follows simply from the linearity of expectation, but I am not convinced. Suppose it were the case that the expectation for a single segment of length $\ell$ was given by $\ell^2$. Then if $X$ and $Y$ are the random variables representing the two needles making up the wire we would have $E(X+Y) = E(X) + E(Y) = \ell_X^2 + \ell_Y^2$, and this is not a function of $\ell_X + \ell_Y$ (though it is a function of $\ell_X$ and $\ell_Y$). Of course we secretly know that $E(X) = C \ell_X$, but my goal in asking this question is to prove this fact without actually calculating anything.