Yeah, what matters is the contents of the initial set of families over which we determine probability.
"A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?"
If the man is randomly chosen from the set of all families the answer is 1/2.
If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27.
The reason for the difference is that a boy/girl family has a 1/7 chance that the boy was born on a Tuesday whereas the boy/boy family has only a 13/49 chance.
I agree but set B is not a uniformly chosen subset of A in this case. That is the core of the trick. The rule for choosing B is intuitively uniform but actually slightly favours families with a girl and a boy over those with two boys.
"A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?"
If the man is randomly chosen from the set of all families the answer is 1/2.
If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27.
The reason for the difference is that a boy/girl family has a 1/7 chance that the boy was born on a Tuesday whereas the boy/boy family has only a 13/49 chance.
B G (7/49 probability of a Tuesday boy)
G B (7/49 probability of a Tuesday boy)
B B (13/49 probability of a Tuesday boy)
13 / (7 + 7 + 13) = 13/27