  Initially motivated by their relevance in foundations of quantum mechanics
and more recently by their applications in different contexts of quantum
information science, violations of Bell inequalities have been extensively
studied during the last years. In particular, an important effort has been made
in order to quantify such Bell violations. Probabilistic techniques have been
heavily used in this context with two different purposes. First, to quantify
how common the phenomenon of Bell violations is; and secondly, to find large
Bell violations in order to better understand the possibilities and limitations
of this phenomenon. However, the strong mathematical content of these results
has discouraged some of the potentially interested readers. The aim of the
present work is to review some of the recent results in this direction by
focusing on the main ideas and removing most of the technical details, to make
the previous study more accessible to a wide audience.
