  Intrinsic quantum randomness is produced when a projective measurement on a
given basis is implemented on a pure state that is not an element of the basis.
The prepared state and implemented measurement are perfectly known, yet the
measured result cannot be deterministically predicted. In realistic situations,
however, measurements and state preparation are always noisy, which introduces
a component of stochasticity in the outputs that is not a consequence of the
intrinsic randomness of quantum theory. Operationally, this stochasticity is
modelled through classical or quantum correlations with an eavesdropper, Eve,
whose goal is to make the best guess about the outcomes produced in the
experiment. In this work, we study Eve's maximum guessing probability when she
is allowed to have correlations with, both, the state and the measurement. We
show that, unlike the case of projective measurements (as it was already known)
or pure states (as we prove), in the setting of generalized measurements and
mixed states, Eve's guessing probability differs depending on whether she can
prepare classically or quantumly correlated strategies.
