The simplest way to run an infinite fair lottery?

I've posted two ways to run an infinite fair lottery (this and this). There is also a very simple way. Just take infinitely many people and have them each independently toss an indeterministic fair coin. If you're lucky enough that exactly one person rolls heads, that's the winner. Otherwise, the lottery counts as a failure. The probability of failure is high—it's one—but nonetheless success should be causally possible. And if you succeed, you've got what is intuitively an infinite fair lottery.

My earlier thought experiments requires a version of the Axiom of Choice. This version doesn't, but the earlier ones has the merit of working always or almost always. However, for the purposes of generating paradoxes and supporting causal finitism this version might be good enough.

A note to fellow mathematicians: Any mathematician reading this and some of my other posts on infinite fair lotteries is apt to be frustrated. There is a lot that isn't rigorous here. But I'm not doing mathematics. One can perhaps best think of what I'm doing as a physicsy thought experiment. When I think of independent indeterministic coin flips, take these as actual causally-independent physical processes, e.g., each indeterministic coin flip happening in a different island universe of an infinite multiverse. I am fully aware, for instance, that the stuff I say in this post isn't fully modeled by the standard Kolmogorovian probability theory. For instance, an infinite sequence of i.i.d.r.v.'s X_n with P(X_n=1)=P(X_n=0)=1/2 need not have any possible state such that exactly one of the variables is 1, depending on how the i.i.d.r.v.'s are constructed. That's an artifact of the fact that probabilistic independence as normally defined is not a sufficient model of genuine causal independence (see here). I am also assuming that permutation symmetries in the space of coin flips persist even when we consider nonmeasurable or null sets. Again that's going beyond the mathematics, but justified as a physicsy thought experiment. If we put each coin flip in a relevantly similar separate universe of a multiverse, then of course everything should be intuitively invariant under permutations of the coins. Probabilities understood vaguely as measures of rational believability go beyond the mathematical theory of probability.