Not a given in the sense that it’s not a 100% certainty, that I agree with. But it’s effectively certain in the sense that the probability of observing a violation can be made arbitrarily small, with a suitable choice of numbers.
Or, in other words, consider the following. You let a ball drop a hundred (a thousand, a million…) times. It always falls down. From this, you formulate a law: stuff falls down.
Now suppose that whatever deity has created the universe has made it so that the actual law is: stuff falls down, except once every sextillion times, when it just hovers in place.
You’ve got no data to support that the law is actually the latter, and never will observe any. Thus, you still formulate the law as ‘stuff falls down’. You’re completely justified in doing so; however, you happen to be wrong.
That’s the situation we’re in here: whenever we try, we will find greyness increasing with overwhelming probability. That is, in any concrete, reasonably large series of trials, we won’t observe a violation. Thus, we formulate a law to the effect that greyness always increases. This law stands on equal footing with every other physical law: it’s a generalization from finitely many observations.
There’s no missing premise. I have explained how we can say that the probability is 1/6, without having to assume it: because one out of every six possible evolutions of the die from arbitrary initial conditions ends with it showing any given number.
Take a coin. Suppose you can only throw it in two different ways—way A and way B. Way A always lands heads up; way B always lands tails up. You don’t have control over whether you’ve thrown it according to way A or way B (random initial conditions, you remember). Then, the probability that it comes up heads is 50% on each throw. No further assumptions necessary.
Suppose now you can throw the coin in 20 different ways, 10 of which come up heads, 10 of which come up tails. This yields the same conclusion. As does supposing that there are 100, or 1000, and so on different ways. What matters is that from the set of possible initial conditions, half of them yield heads, and half of them yield tails.
Yes, this is basically right. And for a large enough number of marbles, what’ll happen is that the observer will observe a transition to uniform grey in every case they do the experiment, since the likelihood of the colors separating will decrease with the number of marbles. Thus, at some point, if they do the experiment 10, or a hundred, or a 1000, or a billion times, they won’t observe a violation of the law ‘the system tends to greyness’ with any appreciable probability. Agreed?
This is in contradiction to the system as I set it up. The position at every time step is not random; rather, it is determined (exactly and absolutely) by the microstate corresponding to the position of the previous time step.
You assume that entropy gain and loss are equally likely. You get out that entropy gain and loss are equally likely. This isn’t surprising.