  We introduce a geometrically natural probability measure on the group of all
M\"obius transformations of the circle. Our aim is to study "random" groups of
M\"obius transformations, and in particular random two-generator groups. By
this we mean groups where the generators are selected randomly. The probability
measure in effect establishes an isomorphism between random $n$-generators
groups and collections of $n$ random pairs of arcs on the circle. Our aim is to
estimate the likely-hood that such a random group is discrete, calculate the
expectation of their associated parameters, geometry and topology, and to test
the effectiveness of tests for discreteness such as J{\o}rgensen's inequality.
