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There are different levels of impossible.
Imagine a universe with an infinite number of identical rooms, each of which contains a single human. Each room is numbered outside: 1, 2, 3, ...
The probability of you being in the first 100 rooms is 0 - if you ever have to make an expected utility calculation, you shouldn't even consider that chance. On the other hand, it is definitely possible in the sense that some people are in those first 100 rooms.
If you consider the probability of you being in room Q, this probability is also 0. However, it (intuitively) feels "more" impossible.
I don't really think this line of thought leads anywhere interesting, but it definitely violated my intuitions.
Your math has some problems. Note that, if p(X=x) = 0 for all x, then the sum over X is also zero. But if you're in a room, then by definition you have sampled from the set of rooms- the probability of selecting a room is one. Since the probability of selecting 'any room from the set of rooms' is both zero and one, we have established a contradiction, so the problem is ill-posed.