  We construct new geometric realizations of simplicial and pre-simplicial sets
where the standard $n$-simplex, viewed as the space of probability measures on
$n+1$ elements, is replaced by the space of $(n+1)$-valued random variables,
with the topology of probability convergence. We prove that the map which
associates to a random variable its probability law is an homotopy equivalence
from these new geometric realizations to the classical ones. Finally, we prove
that this realization provides a new Quillen equivalence between simplicial
sets and topological spaces.
