I agree that more information would help the beauty, but I'm more interested in the issue of whether or not the question, as stated, is ill-posed.

One of the Bayesian vs. frequentist examples that I found most interesting was the case of the coin with unknown bias -- a Bayesian would say it has 50% chance of coming up heads, but a frequentist would refuse to assign a probability. I was wondering if perhaps this is an analogous case for Bayesians.

That wouldn't necessarily mean anything is wrong with Bayesianism. Everyone has to draw the line somewhere, and it's good to know where.