  Busch's theorem deriving the standard quantum probability rule can be
regarded as a more general form of Gleason's theorem. Here we show that a
further generalisation is possible by reducing the number of quantum postulates
used by Busch. We do not assume that the positive measurement outcome operators
are effects or that they form a probability operator measure. We derive a more
general probability rule from which the standard rule can be obtained from the
normal laws of probability when there is no measurement outcome information
available, without the need for further quantum postulates. Our general
probability rule has prediction-retrodiction symmetry and we show how it may be
applied in quantum communications and in retrodictive quantum theory.
