  Employing two models, we show that various counting functions of a random
variable defined by restriction or contraction of a ranked set with
multiplicity (e.g., classical and arithmetic matroids) have expectations given
by the corresponding multivariate Tutte polynomial. The first model is based on
a generalization of a convolution formula of Kung (2010), extending from
matroids to ranked sets with multiplicities. This model enables us to compute
the expectations of many familiar polynomials, such as the chromatic, flow and
Ehrhart polynomials, generalizing the classical results of Welsh (1996) on
random graphs. The second model is designed to compute the expectations of
invariants that are generally not evaluations of the polynomials mentioned
above, such as the number of connected components of an intersection of
hypersurfaces in an abelian Lie group arrangement, and the number of lattice
points in a half-open zonotope. In particular, both models yield new
probabilistic interpretations of the arithmetic Tutte polynomial and $G$-Tutte
polynomial. A simple, but seems to be new convolution-like formula for the
Ehrhart polynomials of lattice zonotopes will also be given.
