[stochastimus]

I don’t think it’s necessary to invoke epistemic probability to explain why the original question is tricky. At least for me, it helps to notice that if the question were rephrased “the older child is a boy, what is the probability that the younger child is also?”, then the answer would be the intuitive 1/2. You get into this counter-intuitive situation with the question as asked because of the “at least” phrasing - the remaining possibilities in the contingency table can still apply to one, the other, or both children.

[quickthrower2]

If someone has at least one boy, it is 1/3

If they then say, I have a secret child-ordering X, such that the first in this ordering is a boy, I guess it is now 1/2

However the receiver of 1/3 case knows there exists a secret child-ordering where this is the case.

Weird!

[quickthrower2]

I had a think about this since, and it must have everything to do with experimental setup, and the existence of the observer in the experiment.

Let's set it up like this: You have 4 dads. Each dad has 2 kids covering all the combinations of kids: MM, MF, FM, FF.

Now if you define the experiment as: you are talking to a dad who has one boy. This was set up by the 4 dads being in the playground, and you walk you to one of them randomly and they tell you this. Now in such an experiment there is a 1 in 4 chance that you lead to a contradiction. Therefore in the 4 alternative universes we disband that option, leading to 3 universes. All equally likely. And in one of them the dad has 2 boys.

If the dad tells you the older kid is a boy then you have some info. What info is that? Well it depends on the precondition under which he told you that. At least a couple of options.

(a) He was going to always reveal to you a kid that was a boy, then tell you if it was the older or younger kid. It happens to be the older one this time.

(b) He was going to tell you the gender of the oldest kid.

In version (a) there are no universes discarded, so you are still at 1/3

In version (b) the universe where the older kid is not a boy has been discarded. There are now 2 universes in which you exist so it is 1/2.