[I have also posted this question on mathoverflow]
The Problem of the Mad King's Draft:
Suppose there is country which is ruled by a king who can be either 'mad' or 'normal.' The king rules a a large country with a continuum of citizens who have names j from [0,1]. The citizens do not know whether the king is mad or not but believe both characters are equally likely.
The king drafts citizens. The mad king drafts 2 citizens while the nice king drafts 4 citizens at random, that is, each citizen is 'equally likely' to be drafted. Drafted citizens do not know how many other citizens are drafted.
What is the posterior of a citizen j who is drafted by the king? This is an important question, since drafted citizens want to escape the draft if the king is mad.
It seems the obvious answer is that the correct posterior should be 1/3. But why?
Ideally, I would like to find a general definition of conditional probabilities that applies to the Mad King's Draft and that is analogous to the standard definition of conditional probabilities, found, for example, on Wikipedia(conditional expectation).
None of these standard definitions apply, which is not satisfying.