As I said in my answer, the uncertainty principle is a consequence of the statistical properties of quantum mechanics, so they are related.

On a finite dart board, it is possible to have a constant, nonzero probability density everywhere. Over an infinite area, this is impossible.

@OverLordGoldDragon Yes. However, in this case, the probably density would have to be nonzero everywhere. It is impossible to have a constant, nonzero probability density everywhere while still having the total probably integrate to 1