  In this paper we continue to explore infinitely renormalizable H\'enon maps
with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional
intuition, the Cantor attractor of such a map is non-rigid and the conjugacy
with the one-dimensional Cantor attractor is at most 1/2-H\"older. Another
formulation of this phenomenon is that the scaling structure of the H\'enon
Cantor attractor differs from its one-dimensional counterpart. However, in this
paper we prove that the weight assigned by the canonical invariant measure to
these bad spots tends to zero on microscopic scales. This phenomenon is called
{\it Probabilistic Universality}. It implies, in particular, that the Hausdorff
dimension of the canonical measure is universal. In this way, universality and
rigidity phenomena of one-dimensional dynamics assume a probabilistic nature in
the two-dimensional world.
