  We consider a finite simplicial complex $K$ together with its successive
barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of
a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower
bounds for the expected Betti numbers of those subcomplexes, together with the
average Morse inequalities and expected Euler characteristic.
