  We prove a Gleason-type theorem for the quantum probability rule using frame
functions defined on positive-operator-valued measures (POVMs), as opposed to
the restricted class of orthogonal projection-valued measures used in the
original theorem. The advantage of this method is that it works for
two-dimensional quantum systems (qubits) and even for vector spaces over
rational fields--settings where the standard theorem fails. Furthermore, unlike
the method necessary for proving the original result, the present one is rather
elementary. In the case of a qubit, we investigate similar results for frame
functions defined upon various restricted classes of POVMs. For the so-called
trine measurements, the standard quantum probability rule is again recovered.
