Actually, most of us got it right, although probably not for the right reasons.
Say you meet a man in the street, and ask him "Do you have two children, one of whom is a boy who was born on a Tuesday?" If he says "yes," then the probability he has two boys is indeed 13/27. But if you run across a woman, and she volunteers the information "I have two children, and one of them is a boy who was born on a Tuesday," then the probability she has two boys is 1/2.
The difference is that the man, if he is truthful, must answer "yes" if he has a boy who was born on a Tuesday and a girl who was born on a Thursday. But the woman,being truthful with the same family, has only a 50% chance to make that comment. The other half of the time, she would say "I have two children, and one of them is a girl who was born on a Thursday."
For the man's case, in your probability calculations you must count every possibility where a family has a Tuesday Boy. But in the woman's case, you only count all of cases where the family has two Tuesday Boys. When she has only one, you must count only half of those cases, and the correct answer turns out to be 1/2.
Actually, most of us got it right, although probably not for the right reasons.
Say you meet a man in the street, and ask him "Do you have two children, one of whom is a boy who was born on a Tuesday?" If he says "yes," then the probability he has two boys is indeed 13/27. But if you run across a woman, and she volunteers the information "I have two children, and one of them is a boy who was born on a Tuesday," then the probability she has two boys is 1/2.
The difference is that the man, if he is truthful, must answer "yes" if he has a boy who was born on a Tuesday and a girl who was born on a Thursday. But the woman,being truthful with the same family, has only a 50% chance to make that comment. The other half of the time, she would say "I have two children, and one of them is a girl who was born on a Thursday."
For the man's case, in your probability calculations you must count every possibility where a family has a Tuesday Boy. But in the woman's case, you only count all of cases where the family has two Tuesday Boys. When she has only one, you must count only half of those cases, and the correct answer turns out to be 1/2.
The question you asked is the woman's case.