The important part in the original is that every choice of splitting point has nonzero probability. This means you will _eventually_ land on a splitting number

The important part in the original is that every choice of splitting point has nonzero probability. This means you will _eventually_ land on a splitting number between the two where you will then have 100% chance to be right (instead of 50% chance for all the other times), pushing your average above 50%. If the splitting point is given to you (by an adversary, not an actual random process), this nonzero probability for any given number is not necessarily fulfilled (at least you didn't indicate so).