  The Born rule for probabilities of measurement results is deduced from the
set of five assumptions. The assumptions state that: (a) the state vector fully
determines the probabilities of all measurement results; (b) between
measurements, any quantum system is governed by that part of standard quantum
mechanics, which does not refer to measurements; (c) probabilities of
measurement results obey the rules of the classical theory of probability; (d)
no information transfer is possible without interaction; (e) if two spin-1/2
particles are in the entangled state (sqrt(1-\lambda)|up>|up> +
sqrt(\lambda)|down>|down>), and one of spins is measured by the Stern-Gerlach
apparatus, then the state of the other spin after this measurement will be
either |up> or |down>, corresponding to the measurement result. No one of these
assumptions can be omitted. The method of the derivation is based on Zurek's
idea of environment-assisted invariance [PRL 90, 120404 (2003)], though taken
in rather modified form. Entanglement plays a crucial role in our approach
(like in Zurek's one), so the probabilities are first considered in the case
when the measured quantum system is entangled with some environment.
Probabilities in pure states appear then as a particular case. The method of
the present paper can be applied to both ideal and non-ideal measuring devices,
irrespective to their constructions and principles of operation.
