  Opacity is a general language-theoretic framework in which several security
properties of a system can be expressed. Its parameters are a predicate, given
as a subset of runs of the system, and an observation function, from the set of
runs into a set of observables. The predicate describes secret information in
the system and, in the possibilistic setting, it is opaque if its membership
cannot be inferred from observation.
  In this paper, we propose several notions of quantitative opacity for
probabilistic systems, where the predicate and the observation function are
seen as random variables. Our aim is to measure (i) the probability of opacity
leakage relative to these random variables and (ii) the level of uncertainty
about membership of the predicate inferred from observation. We show how these
measures extend possibilistic opacity, we give algorithms to compute them for
regular secrets and observations, and we apply these computations on several
classical examples. We finally partially investigate the non-deterministic
setting.
