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Does the expectation value indicate how many times state a should occur, state b occur, etc, or is the probability that state a should occur (say 0.4) verified by how many quantum systems from the ensemble take on that definite state (i.e. seperate from verifying the expectation value)?

Does the entanglement of the experimental systems with the 'outside world' (external from the experiment) need to be taken into account, or can we safely ignore them? I ask because if there is entanglement with everything external from the experiment, the external world is constantly changing (forces acting on the Schrodinger equation) so you won't be probing an idential wave function each time (detector entangled to external world -- the detector becomes entangled with the system it is probing as a consequence of the linearity of the equation).