Given a primitive positive integer vector a ? Zn > 0, the largest integer that cannot be represented as a non-negative integer combination of the coefficients of a is called the Frobenius number of a. In a series of papers V.I. Arnold initiated the research to study the average size of Frobenius numbers, and in a recent paper, Bourgain and Sinai showed that the probability of a "large" Frobenius number is "comparable small". Based on an approach using methods from Geometry of Numbers we can strengthen this result in such a way that we can estimate the average size of Frobenius numbers. Together with a discrete version of a reverse arithmetic-geometric-mean inequality by Gluskin and Milman, this allows us to show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound, which, in particular, strengthens a recent result of Marklof on the asymptotic distribution of Frobenius numbers. Furthermore, we discuss generalizations to the case of more than one input vector a.