I believe I already agreed that it was multinomial, though I also said I felt it could be stated as binomial. After all, the former is just a generalization of the latter.

To be honest, I'm not sure you're understanding the problem. This may be my fault for not explaining it clearly. So let me try a slightly different angle.

You suggested thinking about the problem in terms of a multinomial (binomial) -- which I agreed with -- and suggested applying MLE -- which I also agreed with. However, ultimately I'm not interested in just solving for p (probability of success, or probability of "blue" in your example). I’m interested in solving for the parameters of the underlying distribution that drives p.

So yes, I can do as you suggested and establish the likelihood function for a multinomial distribution, take the natural log, take the derivative with respect to p, and solve for the p that maximizes log-likelihood. Then what? I’m left with an estimate of p, which doesn’t get me anywhere because p is a function of mu and sigma. My first thought was to substitute the Normal CDF as p – in other words include mu and sigma in the likelihood function explicitly – but again the CDF has no closed form solution, so this didn’t seem to be a viable approach.

I don’t know if this is coming across, but in any case I don’t think the problem is as easy as you are making it out to be. If I’m mistaken about any of the above, please let me know where.