  In earlier work we introduced geometrically natural probability measures on
the group of all M\"obius transformations in order to study "random" groups of
M\"obius transformations, random surfaces, and in particular random
two-generator groups, that is groups where the generators are selected
randomly, with a view to estimating the likely-hood that such groups are
discrete and then to make calculations of the expectation of their associated
parameters, geometry and topology. In this paper we continue that study and
identify the precise probability that a Fuchsian group generated by two
parabolic M\"obius transformations is discrete, and give estimates for the case
of Kleinian groups generated by a pair of random parabolic elements which we
support with a computational investigation into of the Riley slice as
identified by Bowditch's condition, and establish rigorous bounds.
