If I speak to somebody who doesn't know what "deterministic" or "random" means, I can give a precise, clear, mathematical definition.
No, you can't, really. Saying "Any live cell with fewer than two live neighbours dies with a probability of 0.5" is an example of a rule that defines a random process simply substitutes one undefined term ("random") with another ("probability"). If that's the kind of definition you want, we can just as easily say "Any live cell with fewer than two live neighbours dies if it chooses to" is a definition of free will. In other words, if "random" means "chance", and "determined" means "rule-bound", then "free will" means "choice"-- but all we've done here is to shift the terms: now we have to define "Rule", "Chance", and "Choice". And, as you might imagine, there is no "precise, clear, mathematical definition" for any of those terms, in a philosophical setting.
You can see an overview of some common conceptions of free will here, but you'll notice that none of them come close to the mathematization you desire.
Similarly, efforts to define rule-based behavior fail, for the reasons found in Wittgenstein's Philosophical Investigations (but also intimated in Hume.)
