Probability does not bite; describing partial information in English bites.
It's not actually true that the probability is 1/3, nor that the probability is 1/2. (Same with 13/27 vs 1/2). The problem is underspecified. Here's two different more specified versions for which the answer is clear:
1. Sample from all two-child families with at least one boy. What portion of these families have two boys? (answer, rot13: n guveq)
2. Choose a random two-child family, then knock on their door. A boy answers. What are the odds the other child is a boy? (rot13: bar unys)
These are both consistent with the description "at least one child is a boy"!
The day-of-week versions:
3. Sample from all two-child families with at least one boy born on a Tuesday. The odds both are boys? (nyzbfg unys)
4. Knock on the door of a random two-child family. A boy born on Tuesday answers. Odds both are boys? (n unys)
It's not actually true that the probability is 1/3, nor that the probability is 1/2. (Same with 13/27 vs 1/2). The problem is underspecified. Here's two different more specified versions for which the answer is clear:
1. Sample from all two-child families with at least one boy. What portion of these families have two boys? (answer, rot13: n guveq)
2. Choose a random two-child family, then knock on their door. A boy answers. What are the odds the other child is a boy? (rot13: bar unys)
These are both consistent with the description "at least one child is a boy"!
The day-of-week versions:
3. Sample from all two-child families with at least one boy born on a Tuesday. The odds both are boys? (nyzbfg unys)
4. Knock on the door of a random two-child family. A boy born on Tuesday answers. Odds both are boys? (n unys)