Wait... If I understand correctly, we don’t have to search the entire range—just from 50% and above?
On average, a random value is always on the... average position.
And on average, whatever method you choose will find the key after 50% of all possible tries.
And the average of all possible methods, for any random value, which can be anywhere, is, you guessed it, on the average, that is, in the middle. What's the average of all possible values between 0 and 1? The middle maybe?
I don't think it's really so hard to grasp the idea. This is what averages mean after all.