Consider a system made out of a possibly large number of identical copies of a two-dimensional dispersive billiard table and let us further assume a form of infrequent pairwise energy-preserving interaction among them. The interaction we will consider will typically be of collisional type and may therefore induce the exchange of a substantial amount of energy among the colliding pair. The question we address is the following: What is the spectrum of Lyapunov exponents of such a system? It turns out this question is closely related to a famous problem in probability theory, first addressed by Laplace in his attempt to construct an error function towards the end of the 18th century: What is the distribution of the ordered lengths of a fixed number of random divisions of the unit interval?