I hope another answer is okay. Absolutely nothing of what follows is mine. The source is given at the end. That main concern of that article, however, is de Finetti's theory of subjective probability and Hume's problem of induction.
It has been suggested that an important step for the development of ideas of probability is the notion of what Jerzy Neyman termed Fundamental Probability Sets (FPS): a set of outcomes that are all equally likely. Such sets naturally allow us to assign numbers to every outcome. In fact, this is how the subject is still introduced in school.
Suppose we suppress what we understand about probability, and how we currently often use terms such as equally possible, equally probable and equally likely interchangeably; and consider the following heuristic, it still seems very suggestive:
If all alternative outcomes are equally likely then none of them should obtain.
The idea being that if any one of the alternatives were to obtain it would break the symmetry that led them to be included in the FPS in the first place. (I think this exposes the circularity at the heart of Bernoulli's defining equally probable on the basis of equipossibility, that Reichenbach talks about)
According to G.E.L. Owen (wiki) the above heuristic is a 'very Greek pattern of argument'. It has a name: Ou Mallon. It is employed, for example, by Anaximander to argue that the Earth is fixed at the centre of the universe, by Parmenides to argue that the universe is uncreated.
Thus, even if the Greeks were mathematically proficient, their existing pattern of reasoning blocked them from identifying FPSs and using it to develop notions of probability, since for them 'equally likely' means none of the outcomes should be possible.
A counter view of sorts is voiced in the first chapter of Hacking's brilliant The Emergence of Probability, called an Absent Family of Ideas (this whole chapter is worth reading in connection to your question). While not addressing this particular point, Hacking dismisses the general argument that people in ancient times failed to notice FPSes.
 Zabell, S. L. Symmetry and Its Discontents;
Skyrms, B., & Harper, W. L. (eds). Causation, Chance, and Credence, Vol. I, 155-190. (1988). Kluwer Academic.
P.S.: An interesting flip in perspective seems to happen by the 1700s. The symmetry conditions, that for the Greeks would block an outcome from obtaining, becomes the very grounds for chance to operate.
Consider this passage from Hume, A Treatise of Human Nature, 1739:
Since therefore an entire indifference is essential to chance, no one chance can possibly be superior to another, otherwise than as it is compos’d of a superior number of equal chances. For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the other: That is, in other words, we must allow of a cause, and destroy the supposition of chance; which we had before establish'd. A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another. This truth is not peculiar to my system, but is acknowledg'd by every one, that forms calculations concerning chances.