As the minimum error approaches zero, the probability that the next guess will reduce the minimum error becomes proportional to the minimum error itself - that's because it's like trying to hit a target, and the probability of hitting the target is proportional to the size of the target. This only applies to the error close to zero, because that allows us to treat the probability distribution as essentially flat in that neighborhood, so we don't have to worry about the shape of the curve.
If the next guess does reduce the minimum error, then, on average, it will reduce the minimum error by half. As above, we're treating the probability distribution as essentially flat.
So, we expect that after some number n guesses, the minimum error is reduced by half. We expect that after 2n more guesses, the minimum error is reduced again by half. Assuming this is what happens, then we expect that after 4n more guesses, the minimum error is reduced by half again.
The reduction in error that we're seeing in this imagined playing out is approximately inversely proportional to the number of guesses. The total number of guesses goes from n to n+2n=3n, to n+2n+4n=7n, etc. If we keep going, the total number of guesses becomes 15n, 31n, etc. This approaches a doubling of total guesses. And the error after each approximate doubling is half what it was before.
This is far from a proof. This is crude, fallible reasoning. It's my best estimate, that's all.
A couple of years ago my workplace was running one of those guess-the-number-of-jellybeans-in-the-jar competitions. I don't even like jellybeans all that much, but nonetheless, I held aloft my nonmagic calculator and said "by the power of Galton!" Taking the mean of all the previous guesses, I put that down as my answer. I was out by one bean, and won the jar. I don't think my colleagues have ever been so interested in statistics as they were that afternoon, and I doubt they ever will be again.
I'm going to admit something a bit silly and embarrassing now: that made me feel like a wizard. Not because of the scope of what I'd done, since it was an utterly trivial piece of arithmetic, but because of the reaction it got. I had drawn on arcane lore unknown to my colleagues, and used it to exercise power over the world.
Personally, I think something like solid state semiconductor technology is about as impressive a real-world miracle as one could ever want by way of demonstrating the whole Science Works/Rationality Is Systematised Winning/Maths Has Manifold Real-World Applications thing, but for most people it will never have the impact of intentionally winning a jar full of jellybeans.
So I ask you, LW-readership: what other impressive nonmagical powers do we have, that we can casually demonstrate to everyday people in everyday circumstances?