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We introduce and analyze a quantum analogue of the Law of Excluded Gambling Strategies of Classical Decision theory by the deinition of different kinds of quantum casinos. The necessity of taking into account entanglement (by the wy we give a straightforward generalization of Schmidt's entanglement measure) forces us to adopt the general algebraic language of Quantum Probability Theory whose essential points are reviewed. The Mathematica code of two packages simulating, respectively, classical and quantum gambling is included. The deep link existing between the censorship of winning quantum gambling strategies and the central notion of Quantum Algorithmic Information Theory, namely quantum algorithmic randomness (by the way we introduce and discard the naive noncommutative generalization of the original Kolmogorov definition), is analyzed.
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