Germ-grain models are popular as models for spatial structures, for example in ecology and material science. In a simple germ-grain model, points are distributed over a region according to a Poisson...

Germ-grain models are popular as models for spatial structures, for example in ecology and material science. In a simple germ-grain model, points are distributed over a region according to a Poisson process with intensity λ. With the Poisson points as centers, one places discs (or spheres in R3) with random sizes, independently of each other. One could think of trees of random size located in a forest according to a Poisson process. A simple question is then: what is the probability that a point chosen at random is covered by at least one disc? Find that probability if the radiuses of discs are independent and uniformly distributed between 0 and a > 0.