  Based on a gambling formulation of quantum mechanics, we derive a
Gleason-type theorem that holds for any dimension n of a quantum system, and in
particular for n = 2. The theorem states that the only logically consistent
probability assignments are exactly the ones that are definable as the trace of
the product of a projector and a density matrix operator. In addition, we
detail the reason why dispersion-free probabilities are actually not valid, or
rational, probabilities for quantum mechanics, and hence should be excluded
from consideration.
