Firstly, as presented it is clearly ambiguous. It is intended to be ambiguous, but in such a way that people who are familiar with the original version will get suckered into believing that it's well formed.
Secondly, if presented precisely, the answer usually given is either 13/27 or 1/2, depending on which version.
Finally, this is like the Monty Hall problem all over again. There are people arguing vehemently and without listening at all, demonstrating clearly that they are excellent at missing the point.
In case you're wondering, here's one statement and answer.
Suppose on knock on people's doors and ask - Do you have exactly two children? If they answer no, I move on. If they answer yes I then ask - Is at least one of them a boy born on a Tuesday? If they say no, I move on.
If they look surprised and say "Yes," what is the probability that they have two boys?
Answer: 13/27.
Yes, it really is.
If you replace the second question with "Is at least one of them a boy with red hair, left-handed, plays piano, was born on Tuesday, and has a cracked left upper incisor" then if the answer is "Yes" then the probability of both children being boys is almost exactly 50%.
If, instead, you replace the second question with "Is at least one a boy" then the probability of two boys is 1/3.
Finally, suppose you see a parent that you know has two children in the park with a boy. Now the probability of two boys is 50%, because, assuming uniform probabilities, having two boys makes it more likely you see them with a boy.
tl;dr: It's hard, and depends precisely on the assumptions you make.
It then goes on to acknowledge that the 33% answer would be correct if we specifically choose a two child family that fits the parameters of the problem beforehand.
It's all about why the information was selected.
Personally, when I read that I could spot the error. For lazy people (tl;dr types) ... order doesn't matter as you're not given any info about that order.
So given 2 children, there are only 3 possibilities ...
boy, boy
girl, girl
boy, girl
So if you're told one of them is a boy ...
boy, girl
boy, boy
It's pretty clear I won't convince you.
Let me leave you with these questions:
If I toss two coins until at least one shows a Head, what's the probability that both are Heads?
If I roll two dice until at least one shows a 6, what's the probability that both are 6's?
If I spin two roulette wheels until at least one shows a Red-23, what's the probability that both are red?
Are you sure?
Older child boy, younger child boy
Older child boy, younger child girl
Older child girl, younger child boy
Older child girl, younger child girl
conditioning upon more events can lead to a higher probability. p(boy = 2 | boy >= 1) < p(boy = 2 | boy >= 1, tuesday) (or more precisely, conditioning upon more events can yield a distribution with less entropy)
p(boy = 2 | boy >= 1) = p(any child = boy)
you're right, without this explicit construction, it's problematic.
The statement "I have two children, one of whom is a son born on a Tuesday" is semantically ambiguous. It can mean (1) "I have two children, and the quantity of them who are males born on a Tuesday is exactly one", (2) "I have two children, at least one of whom is a male born on a Tuesday", or even (3) "I have two children, and the maximum quantity of males born on the same Tuesday is one".
The problem I have is attempting to assign a probability to something that isn't reasonably known to be based on randomness. Did the questioner arrive at this statement by picking an arbitrary child and then declaring his gender and day of birth? Or did he pick his favorite child and declare his gender and day of birth? Would he have used this same question if they had both been born on Tuesday, or would have have picked a different distinguishing feature?