Which type? Algorithmic? Algebraic? Measure-theoretic? Combinatorial? Quantum mechanical? Set-theoretic? Nonstochastic? ML-randomness? Irreducible? Decision-theoretic? Or did you really believe that there is some agreed upon definition of random that can serve as a foundation to your argument?
You absolutely would.
1) You wouldn't
2) 1/0 is undefined
3) There is always a chance to determine a probability
4) Probabilities of 0 don't mean impossible and probabilities of 1 don't mean certain (hence the change in terminology from measure theory/integration theory in general such as a.e (almost everywhere) to a.s. or "almost surely").
5) Randomness isn't required for improbable or impossible, and the two aren't really very related. It is virtually impossible to flip a billion coins and obtain a sequence of all heads, but equally virtually impossible to obtain any other sequence.
6) If you can't determine probabilities, you have no measure of randomness. If you can measure, characterize, or otherwise quantify randomness, then you immediately have the ability to determine probabilities.
7) I'm bored. There are free sources on probability theory, logic, etc., out there, from Harvard courses to free textbooks (I've even gone to the trouble of trying to evaluate the best free sources in these and other areas:
Math Books/Resources for Free and for Learners and in particular see the post
Package 2: Probability & Statistics)