Here is a post at the corner concerning a probability question. It leads to a number of other posts and to Derbyshire's analysis here. If you follow the chain of posts back, it leads to other sites and considerable debate over the interpretation of this problem.
The analysis is tricky only if the problem is interpreted in something other than its straightforward, plain meaning. The statement of the problem is:
"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"
Now the problems all come from that "born on a Tuesday" clause in the middle sentence. Take that out, and everyone agrees on the answer:
"I have two children. One is a boy. What is the probability I have two boys?"
This is the classic coin-flip enumeration problem. Having children is like flipping a coin, with heads = boys and tails = girls. The possible outcomes for two consecutive coin flips are:
Heads - Heads
Heads - Tails
Tails - Heads
Tails - Tails
or, in the boy girl terms:
Boy - Boy
Boy - Girl
Girl - Boy
Girl - Girl
Since we know that at least one of the children is a boy, the last case is ruled out and the probability that the speaker has two boys is one in three.
Returning to the original problem, the analysts all seem to think the phrase "born on a Tuesday" is very significant, but they can't agree on its significance. I don't think it adds anything to the problem at all. In the straightforward, obvious interpretation, it is only a statement after the fact of birth concerning the day of birth. It's like saying the boy was 8 lbs at birth, or was born with blue eyes. It doesn't say anything about the prior possibilities of weight or birthdays; it is only a statement about what in fact occurred. It doesn't say that one or both boys couldn't have been born on a Wednesday. If that had happened, the consequence would be that the problem would say:
"I have two children. One is a boy born on a Wednesday. What is the probability I have two boys?"
The answer to this question is the same as the answer to the Tuesday question and to the simpler question that does not refer to a day at all: 1/3.
Derbyshire calculates the probability as 13/27. He can only get there by interpreting the "Tuesday clause" as affecting the prior probabilities of birth. In other words, the case of two boys born on Wednesday need not be included in our enumeration of cases because it wasn't possible for both boys to be born on Wednesday, since we know one was born on Tuesday! I hope everyone can see the post facto fallacy in this reasoning. Anyone who would buy this line of reasoning is playing the role of the father in the following comic scenario:
One day you get a letter from the town correcting your son's birth certificate. He was born two seconds after midnight so he was actually born on a Wednesday rather than a Tuesday. With a heavy heart, you sit your son down and tell him the unfortunate news: "I'm sorry to tell you this son, but I'm not your real father. My son could only have been born on a Tuesday, and I've just learned you were born on a Wednesday."
By the way, this problem is not comparable to the Monty Hall problem. The Monty Hall problem is a genuinely counter-intuitive probability result. The Tuesday's Child problem is more like a riddle or joke that depends on deceptive or ambiguous language.