When I read this question (problem restated below), and the first comment, I was drawn to the great similarities between this problem and the Monty Hall problem (asking for the winning probability if you switch). Both include a choice with partial information about what's hidden behind the doors / in the compartments. In both cases it's very easy for lay-people to assume the probability is $1/2$, when in fact it's $2/3$.
Is there an easy translation between the two problems? What do the goats and car correspond to in the boxes-with-gold-and-silver-bars problem?
I have some ideas, stated in an answer below, but it's far from complete, and I would like to hear other ideas as well. If there is some abstract, literal translation then I would love to hear it.
I have three boxes, each with two compartments.
One has two gold bars
One has two silver bars
One has one gold bar and one silver bar
You choose a box at random, then open a compartment at random.
If that bar is gold, what is the probability that the other bar in the box is also gold?