The partition Z is just the sum of the state weights, and dividing each weight by Z turns it into a probability that adds up to one. Normalization is bookkeeping, not physics: it converts raw weights into fractions of the whole. The exact result holds only for the listed states and stated weights; real systems need a preparation argument for why those weights apply, plus sampling and model uncertainty.
A finite weighted ensemble has one checked normalizing total. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.