  Betti numbers of configuration spaces of mechanical linkages (known also as
polygon spaces) depend on a large number of parameters -- the lengths of the
bars of the linkage. Motivated by applications in topological robotics,
statistical shape theory and molecular biology, we view these lengths as random
variables and study asymptotic values of the average Betti numbers as the
number of links n tends to infinity. We establish a surprising fact that for a
reasonably ample class of sequences of probability measures the asymptotic
values of the average Betti numbers are independent of the choice of the
measure. The main results of the paper apply to planar linkages as well as for
linkages in R^3. We also prove results about higher moments of Betti numbers.
