Suppose I have a particle whose quantum state is known to be exactly spin-up along the z-axis. Then suppose I measure its spin along the z-axis. Quantum mechanics (QM) states that the probability of measuring spin-up is 100% and that of spin-down is 0%.
Does this 0% mean that measuring spin-down is impossible (forbidden and cannot occur), or does it mean that measuring spin-down is "possible but infinitely unlikely" (similar to, say, the probability of hitting the exact center of a perfect, mathematical dart board)?
On the one hand, some may argue that the question is moot because in practice, we cannot know the exact quantum state of a system nor measure exactly along the z-axis due to experimental limitations. My counter point would be that nothing in the mathematical formalism of QM prevents us from talking about this situation, and so we should expect the mathematical theory to make an exact prediction with a clear physical meaning. The formalism should provide an unambiguous answer for any ideal case, and then we can worry about experimental imprecision as an afterthought.
This point was never raised during any of my QM training, and I have always thought of zero probability as "impossible" in this context. But given that probability is not well understood philosophically, and that there are situations where "X has probability zero" does not imply "X cannot occur," I think this question deserves some thought.