  We study mathematical expectations of Betti numbers of configuration spaces
of planar linkages, viewing the lengths of the bars of the linkage as random
variables. Our main result gives an explicit asymptotic formulae for these
mathematical expectations for two distinct probability measures describing the
statistics of the length vectors when the number of links tends to infinity. In
the proof we use a combination of geometric and analytic tools. The average
Betti numbers are expressed in terms of volumes of intersections of a simplex
with certain half-spaces.
