The flaw with assuming an event will happen inevitably because it is not zero makes the assumption that events are created equal and that there is diminishing probability of the event not happening. I argue that certain events, just because they would not necessarily never happen, still may never happen anyway. Also, said events could occur infinite times because a fraction of infinity is infinity (and the law of large numbers) (calculus makes understanding this last tenet easier). Warning - large wall of text - maybe skip to conclusion to avoid numerous details unless you wish to respond to my conclusions, which is in a light blue color.
1. Coin flipping is an example of this situation in favor of saying an event (a tails) not guaranteed not to happen will LIKELY eventually happen, but may not actually ever happen. The odds of the i-th flip of the coin being the first tails flipped in a sequence is (1/2^i), because (for i = 5) the odds of 4 heads in a row is 1/16 (2^4) and the odds of a tails the next flip are 1/2, and the events are independent, so 1/16 * 1/2 = 1/32. Thus it is POSSIBLE that no tails is ever flipped, but the odds as we flip infinitely many times becomes lim(n-> infinity) [1/(2^n)] = 0, so mathematically if you could flip it an infinite number of times the odds of all heads is zero, except that it's impossible to flip it an infinite number of times and so there is always a small positive chance, although I'd put my money on the calculus of limits and say no, there will be a tails eventually (assuming a fair coin and fair flip).
2. Some events are not as simple as a coin flip, hence the odds happening vary wildly, from it being possible/guaranteed to it never happening. I'll go with the extreme example to make it easy - nuclear war. Yes, the odds of a nuclear war happening are non-zero (a nuke could be launched and more could follow, in a war). But there are certain events that must occur that are NOT a "flip of the coin" so to speak that make the odds so close to zero that it is arguable (through limit calculus) the realistic odds are zero. For instance, there would need to be a situation where the benefits of launching nuclear weapons outweighted (almost surely) inevitable retaliation. I will make some assumptions here that you may take issue with, but given retaliation odds of 100%, and odds near zero that a state will become so totalitarian and vicious that it will either deploy nukes or not accept unconditional masurrender, a rational human being will not take the risk of launching nuclear weapons. Nuclear weapons could belong to fundamentally irrational people, but (assumption you can critique) those people chose to engage in acts of terrorism rather than look to work through the state, and their assaults on the state would (another assumption) result in multilateral national backlash, precluding actual dominance by an irrational power. Thus, the barricades to launching nuclear weapons (I've identified 2), are so astronomically difficult to surmount, that the odds of a nuclear weapon being launched, even as time goes to infinity, are so vanishingly small that I would believe application of L'Hospital's rule (for dealing with 0 * inifinity limts) would still yield odds of zero, as the odds of nuclear weapons being used as the variable e^t (another assertion), t being a barrier, probably approaches zero far faster than the growth of time t to infinity (ie it is like lim(t-> infinity) [t/e^t] which = 0), thus it is possible that it could happen but never actually should happen.
3. Events like a flipping a coin could occur infinitely often given infinite time (and infinitely many heads could occur) as the law of large numbers states that as you engage in an event more and more times with specific, mutually exclusive outcomes, the percentages of times those events occur becomes closer to the theorhetical percentages (calculated with probability and calculus as it goes to infinity). Thus as you flip a coin infinitely many times, you should end up with ~ 50% heads and ~ 50% tails, and using limit calculus, infinitely many flips (n) divided in half means there are lim(n-> infinity)[n/2] = infinitely many heads and also infinitely many tails, since this calculates each.
In conclusion, I think it is fair to state that A) given real-world scenarios, it is possible but just barely so, to have an event never occur, but the odds are so minute as to be laughable and about 0, B) Events may have nonzero probability and still not occur because the odds are so small they "overpower" the infinintely many attempts via L'Hospital's rule, which is for when we have lim(n-> infinity) [(quantity rapidly approaching zero) * (quantity approaching infinity)], or in other words, that the odds are functionally zero, and C) that even mutually exclusive events could each occur infinitely many times due to how infinity operates (both heads and tails can be flipped infinitely many times when there are infinitely many flips, because the probability is of either result is not zero). Note that this is just how I think the math works out. Maybe that's not what you wanted though.
Essentially, the odds of some events not happening are zero (they will occur), but I think that only applies for events that have even or fixed ratios of the predicted outcomes (3:1, 1:1, etc.) and independent of other variables - the variables in my example (point 2) stack up so damagingly (and would increase over time (in my opinion) as aversion to war seems to go up as a function of time as well) that the odds can stay zero of the example war. Given even odds of multiple different outcomes, each outcome could occur infinitely many times (limit calculus)
