We're far from understanding everything about the brain but we know a lot and have good reason to think that everything we do is the product of neuron firings.

Neurologically speaking, I'll admit that you could be right. It could turn out to be the case that humans happen to be a sort of creature that makes decisions based on neuron firings, and that neuron firings are in turn based entirely on deterministic particle collisions plus a bit of quantum randomness on the side. I keep half an eye on the neuroscience articles in the popular press, and if and when they report that conclusion, I'll take it seriously. It would force me to revise several of my core beliefs. One belief that wouldn't change, though, is the belief that, as a matter of philosophy, there's nothing wrong with the idea of a choosing being, even if we have no real life examples of choosing beings on Earth.

Your ignorance is not cause for postulating fundamental ontologies. It is a mysterious answer. If we knew more about ethics and psychology you absolutely could do this calculation.

I've read and re-read the page on mysterious answers, and I think it's a great article. I don't mean to shut off inquiry at all by saying that I choose things; by all means, scan my brain and tell me what you see! I'll be mildly curious. I'm not curious enough to do the scans myself, because I'm too busy trying to investigate my mind at the level of macrophenomena like "habits," "willpower," and "awareness" to bother much with the microfoundations of those phenomena in individual neurons or clusters. I expect that within my lifetime, brain scans will be cheap, safe, and precise enough that I'll be able to get better information from physical science than from introspection, and then I'll switch the bulk of my investigative activity over to physical science. In the meantime, I find that "choice" works just fine as a placeholder in the heuristic equations I use to model my mental macrophenomena. It may or may not correspond to anything real at the quantum level, but it helps me understand myself, so I'm using it.

Also please distinguish choice from randomness. I'm still not quite sure what this concept consists of. How can something be uncaused but not random? If there is no cause what constrains the outcome? If there is no constraint on the outcome how is it not random?

I'm sorry; I'm not sure what else to say. if my analogy about red/blue/green and my necessary vs. sufficient paragraph didn't get the point across, then I don't know how else to explain it. if you insist on using a mental model that has only two possible values for a variable, then talk about a third value will not make any sense to you; I cannot stop you from trying to explain the third value in terms of your existing mental model and then getting confused or annoyed when it doesn't work.