  We consider the probabilistic applicative bisimilarity (PAB), a coinductive
relation comparing the applicative behaviour of probabilistic untyped lambda
terms according to a specific operational semantics. This notion has been
studied with respect to the two standard parameter passing policies,
call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy not
reducing within the body of a function. In particular, PAB has been proven to
be fully abstract with respect to the contextual equivalence in cbv but not in
lazy cbn. We overcome this issue of cbn by relaxing the laziness constraint: we
prove that PAB is fully abstract with respect to the standard head reduction
contextual equivalence. Our proof is based on the Leventis Separation Theorem,
using probabilistic Nakajima trees as a tree-like representation of the
contextual equivalence classes. Finally, we prove also that the inequality full
abstraction fails, showing that the probabilistic applicative similarity is
strictly contained in the contextual preorder.
