  We study bisimulation and context equivalence in a probabilistic
$\lambda$-calculus. The contributions of this paper are threefold. Firstly we
show a technique for proving congruence of probabilistic applicative
bisimilarity. While the technique follows Howe's method, some of the
technicalities are quite different, relying on non-trivial "disentangling"
properties for sets of real numbers. Secondly we show that, while bisimilarity
is in general strictly finer than context equivalence, coincidence between the
two relations is attained on pure $\lambda$-terms. The resulting equality is
that induced by Levy-Longo trees, generally accepted as the finest extensional
equivalence on pure $\lambda$-terms under a lazy regime. Finally, we derive a
coinductive characterisation of context equivalence on the whole probabilistic
language, via an extension in which terms akin to distributions may appear in
redex position. Another motivation for the extension is that its operational
semantics allows us to experiment with a different congruence technique, namely
that of logical bisimilarity.
