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.
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