Probability Problem

There is an interesting probability problem floating around the net. It is easily stated:
"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

The "convential wisdom," from the math experts, is that the answer is 13/27th, not the 1/2 that one might expect. I read about this problem here and here and here, the last link being to John Derbyshire's blog. He has comments in the corner too, here, here, and here.

I came up with the same solution, but later had my doubts. I now side with the folks who say the answer is 1/2. The problem is one of language, which is ambiguous.

When someone says "I have two children. One is a boy born on a Tuesday" does he mean that he has at least one Tuesday-born boy, or does he mean that one specific kid, call him William, was born on a Tuesday? If the former, the answer is 13/27. If the latter, the answer is 1/2. That's the ambiguity.

Consider another, simpler, scenario: "I have two children. One is a boy." Has he told you that at least one of his kids is a boy? If so, then the probability of two boys is 1/3. Has he told you the gender of one specific kid chosen at random? If so, the two-boy probability is 1/2. What do you think is the more common meaning of the statement? I feel pretty sure that the latter interpretation, yielding 1/2, is far more often correct. This is not the interpretation of the "experts" linked above, who would give the answer as 1/3.

A comment posted here agrees with me and is a good explanation. Another great comment here.