Q&A: A Die and Randomness
A Die and Randomness
Question
You wrote somewhere, and I don’t remember where, though I think the claim is agreed upon, that in rolling a fair die there is no randomness at all, and yet the distribution of outcomes is one-sixth for each face. The idea is explained by the high sensitivity to the initial conditions (for example, the exact angle of the throw).
So then I didn’t understand why it really comes out one-sixth. Is this a stupid question? Is it my free choice, without being aware of which angle to pick, that produces the real randomness here?
Answer
Yes. There is very little chance that two choices of initial conditions will land on exactly the same angle and force.
Discussion on Answer
I don’t understand what is unclear. These things are connected to ergodic theory, but intuitively it’s completely obvious. This isn’t the place to spell it out.
If I’m choosing within the space of initial conditions, then I understand both the randomness and the one-sixth. But if determinism goes to some particular point there within the set of initial conditions (let’s imagine the space of initial conditions, say there are only two conditions, angle and force, so it’s a plane where each pixel is colored in one of 6 colors), then why do a thousand deterministic throws come out one-sixth, one-sixth? I’m probably missing something simple here. Why should this determinism care about probability? If you tell me that a thousand deterministic throws fall on a thousand pixels inside a small square, and that square is itself also distributed one-sixth, one-sixth, then fine. But determinism doesn’t take a whole region; it just goes to points according to what happened before, so why should it get to one-sixth? Actually, you probably already answered, but I’m not getting it. I probably just wrote nonsense.