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Based on a gambling formulation of quantum mechanics, we derive aGleason-type theorem that holds for any dimension n of a quantum system, and inparticular for n = 2. The theorem states that the only logically consistentprobability assignments are exactly the ones that are definable as the trace ofthe product of a projector and a density matrix operator. In addition, wedetail the reason why dispersion-free probabilities are actually not valid, orrational, probabilities for quantum mechanics, and hence should be excludedfrom consideration.

A Gleason-type theorem for any dimension based on a gambling formulation of Quantum Mechanics