It began to rain as I was standing outside today, and while watching the raindrops wet the cement I thought of the following question:

Assuming $n$ raindrops (i.e. circles of radius $r$) fall randomly, independently and uniformly over a square of area $A$, what is the expected area of the wet region?

As $r/A$ approaches zero, the expected area should approach $\pi r^2 n$, but when the raindrops aren't tiny, the probability of drops intersecting isn't negligible. Also, there are edge effects because the square is finite, but I'm not so interested in those. Actually, the problem could be generalized and restated as follows:

If $n$ circles of radius $r$ are drawn at random in the plane such that their centers lie in some region $S$ (we could take $S$ to be a square), what is the expected area of their union?