Do you mean randomness in a pure sense, or in a computational sense, like pseudo-randomness? In terms of the pure sense, I'm not convinced pure randomness exists, so there's your answer right there. In terms of some pseudo-random computation over Buchi-automatons? You would have to parse the omega-regular inputs in such a way that the algorithm could apply, and such an answer would have a result in relationship to the asymptotic density of the string.

I share your skepticism about randomness (I think that Leibniz was on to something), But for purposes of this exploratory question, let's keep it simple: a Random selection of some event N is the Probability of its selection from all possible outcomes under consideration. So, for instance, the probability of randomly selecting the number 51 from the set of the first 100 Natural Numbers is 1/100 or P=1%, Obviously, as the number of possible instances increases in the denominator, P asymptotically approaches 0 at infinity. Hence, in an infinite set P would be 0.

I do think that it is a non-zero, infinitesimal probability, and in the context of theoretical computation, it's measure is related to the language one chooses to represent the set when performing the probability algorithm, which we will assume exists as a blackbox. I do not know how this will work with an uncountable system, but for a countable transfinite system, consider the following:

Let us work with the natural numbers for simplicity sake. Let us represent the natural numbers, in order, on a linear tape, to be read by some Buchi automaton, each with a binary representation. Each number will be preceded by a marker symbol, #, which indicates the beginning of a new number, such that the tape will begin and proceed as the following: {#1#10#11#100#101...}.

Make a Random selection upon a non-# such that the selection will yield either 0 or 1. The probability that one will select a 1 is greater than the probability that one will select a 0, since the asymptotic density of the string is greater with 1 than with zero by the notational properties of the natural numbers represented in binary (every natural number contains a 1, however, not every natural number contains 0), making those numbers with a representation that includes a 1, far greater than those who don't. From here, we can conclude that the random selection will prefer a number that includes a 1 (100%) and that the random selection from the series of 1s and 0s will also more likely produce a 1 based on the asymptotic density of the string.

Now make a Random selection upon the subset of # symbols. This selection will yield some #, but the question is, what is the probability that the natural number which comes after the # sign, is some specific natural number? And since the asymptotic density of the number of # from all # is 100%, we get an even probability, that is, no one number can be chose more than another. But what does this mean in a transfinite context? And here we must introduce the infinitesimal. To me, I think the selection of some random element with equal probability of selection over a transfinite field, is itself a very good definition of an infinitesimal (I think it should be over a field, although it doesn't have to be), but since we are not using a field with the natural numbers, for simplicity's sake, let us just define an infinitesimal, "small," as the inverse of the rank of the natural numbers, such that omega times small is 1. Small is non-zero, or omega times small would be 0, however, small has some zero-like properties, some of which you have already mentioned.

I'm just going to jump without thorough proof here, as I hope you can see it, the probability of your answer is P=small for any #. For any random 1, you get the asymptotic density of 1 on the string times small, which is something like (forgive my impreciseness here, so caveat emptor at these figures): P=log(omega)*small or maybe P=small+small, Anyway, it will be larger than just small.

I hope you don't mind all the original research above, most of this is just off the top of my head right now but loosley based on a paper I recently wrote. I'm still hoping a reviewer will come forward to read it, it is my first paper in theoretical computation and the foundations of mathematics, but my findings might be a bit controversial and I'm a bit of a lone wolf.

PS, If you are looking for a real number answer, I don't think it exists. Zero isn't a sufficient answer, and the Real numbers are not powerful enough to describe a rank over the Naturals. If we are limited to the Reals, the answer is undecidable.

Could you please elaborate on what you mean specifically in asserting that "if we are limited to the reals, the answer is undecidable"? What answer exactly? Undecidability in what sense? I'm assuming that you're accepting CT as true.

I do accept CT is true. However, my point in the statement "if we are limited to the reals, the answer is undecidable," has more to do with the existence of the rank of a well-founded transfinite set, which is not a real number, it is a "hyperreal." So the answer to your question, which can be reduced to the probability of some random finite choice over an infinite space, invokes the hyperreal numbers, since we need the rank of that space to formally describe the problem in the first place. So, if we limit ourselves strictly to the Real numbers, the Real numbers do not contain the hyperreals and are not powerful enough to describe your probability type, which will be some infinitesimal, or inverse on the Rank of the set (assuming well-foundedness, which itself is not a property of the Reals). All this is why a Real number answer to your question is just not possible. It doesn't mean that the problem is completely undecidable in the CT sense, since at that moment, when invoking a Turing Machine, we have the issue of binary representation, and we can assign a register or memo which handles the permutations over the hyperreals.

To summarize my thinking here: I accept CT is true. But, if we limit ourselves to the Real numbers (in the sense of pure mathematics), the Real numbers do not include the Rank of the natural numbers, omega, however, that does not mean that computations cannot be registered in such a way these answers themselves are not computable by some representation of the omega or the infinitesimal as some algebra (I hope you see that I'm trying to separate the condition of a Real number from the condition of non-real representation by the Reals). So, undecidable in a very limited sense. Limited to the Reals.

In finite terms, the rank of the set S={1, 2, 3... 99, 100} is 101, and we can actually find the probability of a random selection to any element in a set by the rank of the cardinality of that set.You can also find it in terms of it's supremum, as in your illustration, but there is no supremum in an increasing infinite set such as the natural numbers, so it makes more sense to define probability over an infinite space in terms of the rank of the set. I wish I had more knowledge in probability to describe the terms more accurately for you there, but I'm mostly limited to logic, set theory and computation.

I like your approach about transferring our discourse to a Probability Space, but I'd like to stay on point here (pun intended). Of course a Transfinite Set has no supremum per se, but the heuristic of my question is how a probability selection is affected by the nature of how we chose to characterize the constituents of the line under analysis. In this context, it is my suspicion that a lush flourishing of insights might be gained when we consider not the Reals (or Hyper-reals, for that matter), but the underlying subtext of denumerability and nondenumerability within the universe of discourse and the sets under consideration. A tangentially related aspect of this may be some probabilistic schemes in testing Semantic Security in a hypothetically unbounded encryption algorithm.