  In this article we give a classification of the binary, simple,
$\omega$-categorical structures with SU-rank 1 and trivial pregeometry. This is
done both by showing that they satisfy certain extension properties, but also
by noting that they may be approximated by the almost sure theory of some sets
of finite structures equipped with a probability measure. This study give
results about general almost sure theories, but also considers certain
attributes which, if they are almost surely true, generate almost sure theories
with very specific properties such as $\omega$-stability or strong minimality.
