[cbr]

I don't see how knowing one of the children is named "joseph" gives us any more information than knowing one of them is a boy [1]. So I would be curious to see your logic for why the two have different answers.

[1] Unless you're leaving math-problem-world and want us to consider the probability that a child named 'joseph' is female.

[Peaker]

Let's say we have a village with just 4 families, 1 for each possible case. The names of the boys are b0,b1,b2,b3 and the names of the girls are g0,g1,g2,g3 (as no two names are the same):

b0, b1

b2, g0

g1, b3

g2, g3

If we draw a random family that has at least one boy, we get all the families except the last one. And 2/3 of those have a girl in them, as opposed to 1/3rd that doesn't. So its 66% to find a girl.

However, if I ask you about a particular boy, I could ask about b0, b1, b2 or b3.

b0: sibling is b1

b1: sibling is b0

b2: sibling is g0

b3: sibling is g1

So for half of the possible names I choose, the sibling is a boy, and for half it is a girl.

Assuming the name was chosen at random from the existing names of the boys in the village, that makes the chance 50% to find a sibling girl again.