If you like, you can work out the maximum number of degrees of freedom that could theoretically be involved in something you could call a fair coin toss.
Hint: it’s a free body under rotation on a parabolic trajectory. You won’t end up needing Tony Stark hardware for this one.
Bonus factoid before bed: coins flipped by humans empirically have a small (~51%) bias toward heads. Supposedly folks tend to start with heads up more often, and the coin tends to flip over 0 times more often than you’d think, and that adds up to a bias you can measure in an afternoon.
You’re talking about something different from algorithmic randomness, which would be a property of (infinite...) sequences. When I throw a coin in the air and say “heads or tails”, you have to make a prediction as to where it will land based on the information you have. Not someone else’s information— _your_ information. If you feel like you have as good a chance of winning as losing, you’ll tell us you think it’s a fair — a “random” — coin toss.
You’re talking about betting behavior. It’s not a wild tangent the way some would think — from this we get the structure of probability theory, we get derived preferences, we get the entire mechanics of Bayesian inference that underlies all systems which learn.
From studying this math — the math of the coin flip with a dollar on the table, the math of the universe — we learn that there _cannot be_ “chances” or “odds” out there in the world that we can see if we look hard enough— what we see if we look closely enough will be the structure of our perception, which is a perception itself constituted by subjective probabilities.
We learn that belief states must be subjective, must be local. There can’t be a fact of the matter in any way that makes sense to anyone. And yet we also can prove that we cannot truly reserve judgment, that we cannot be true skeptics— unless we are also not anything recognizable as scientists.
So, yes, absolutely. In the consensus reality we all agree to occupy, how you decide to answer the question “heads or tails” matters — in the deepest possible sense — just as much as anything else. And folks can get _so_ wrapped up in the idea of what an imaginary computer might do, they forget to notice what they are doing with every waking moment.
If you want to get really crazy about it, you can talk about betting about what the 999999999th digit of pi is. It's not random, it's not even really unknown, but I sure don't know the answer without a bunch of extra work.
I haven’t really heard much talk about “Kolmogorov randomness” before, and so I’m wondering if you might be running up against the limits of the Wikipedia paradigm when it comes to pioneering scholarship.
The citation for that paragraph is a peer-reviewed journal article covering Kolmogorov complexity and randomness. It’s actually a really good article, by someone pretty famous named Per Martin-Löf. Which is all great, except that paper is from 1966, and in 2019 a more studied concept is something called “Martin-Löf randomness” :)
Hint: it’s a free body under rotation on a parabolic trajectory. You won’t end up needing Tony Stark hardware for this one.
Bonus factoid before bed: coins flipped by humans empirically have a small (~51%) bias toward heads. Supposedly folks tend to start with heads up more often, and the coin tends to flip over 0 times more often than you’d think, and that adds up to a bias you can measure in an afternoon.