  A finite dimensional quantum system for which the quantum chaos conjecture
applies has eigenstates, which show the same statistical properties than the
column vectors of random orthogonal or unitary matrices. Here, we consider the
different probabilities for obtaining a specific outcome in a projective
measurement, provided the system is in one of its eigenstates. We then give
analytic expressions for the joint probability density for these probabilities,
with respect to the ensemble of random matrices. In the case of the unitary
group, our results can be applied, also, to the phenomenon of universal
conductance fluctuations, where the same mathematical quantities describe
partial conductances in a two-terminal mesoscopic scattering problem with a
finite number of modes in each terminal.
