I’m sorry about that becoming three posts.

In summary, the problem is largely posted because of the seeming paradox when you get 13/27 or 1/3 as the intended (correct) answer. The problem only really works because we don’t normally expect people to give very precise spoken (or written) statements, and so we have a tendency to extrapolate what was actually meant.

If the question had been phrased more like “I have two boys, one was born on a Tuesday, then what is the probability that the OTHER is a boy?”, then the semantics have changed – in fact you have been given more information, because a particular child has been singled out (in my opinion, of course). But (IMO) the original question has been deliberately phrased so as to not be referring to a particular child. That, I believe, is at the heart of why people get confused about the answer.

Of course, some people get confused because they haven’t got a clue about what’s really being asked. When I posted this problem, someone in an attempt to lampoon it, suggested that you may as well ask what if one of the boys had a tentacle growing out of his head? I responded that if the probability that a boy has a tentacle growing out of his head is p, then the probability of there being two boys = (1/2)(1-p/2)/(1-p/4). That varies from 1/3 to 1/2 as p varies from 1 to 0.

On the Tuesday not mentioned version of the problem, a typical attitude is that you know for sure that one child is a boy, so the other being a boy/girl is 50-50. But this would be equivalent to only looking at families where you only (initially) sampled one of the children and didn’t examine the second one if the first was a girl. You’d then be excluding mixed gender families that you shouldn’t have excluded.

Maybe I didn’t say it explicitly, but I too believe that B must equal T. I was emphasizing what happens when they are different, to try to explain why people do not initially expect the 13/27 answer. Essentially (and this comes up a lot in this post), we have no reason to treat the two facts differently, so we must treat them the same. It was for this reason that Gary Foshee’s insists T must be 1, since the accepted answer without “born on” information is that B=1.

But the issue you refer to isn’t quite that “somebody is twice as likely to mention in conversation that they have a son just because they have 2 sons,” even though it reduces to that. It is really two issues. First, are they more or less likely to mention one the gender of one child when they have a mixed-gender family rather than a single-gender family? If they are, there is no way we can answer the question since we don’t know by how much the probabilities differ. Puzzles often leave details like that out, for brevity, and we really can’t assume anything except that these two factors are independent. Second, are they more or less likely to mention a boy this way, than a girl, when they have a mixed gender family? And again, by how much? That’s where my factor B comes in. If the problem doesn’t say or imply a value, we have to assume here, too, that there is no difference, and B=1/2.

The problem is, mathematicians and puzzle designers seem predisposed to believe there is an implied value for this problem, or that there should be and it is B=T=1. Look up Keith Devlin’s (of the blog Devlin’s Angle) response to it last May. He says, essentially, that as worded we really should have used B=T=1/2; but if the problem is worded “given the man has a boy born on a Tuesday,” we should use B=T=1. I think that also is wrong.

The mistake Devlin is making is of taking how we read the expression P(E|C), as “the probability of event E given event C.” He applies a magical implication from the word “given” corresponding to this colloquialism. He overlooks that the expression really means “the probability event E happens, given that the outcome is constrained to event C AND UNCONSTRAINED WITHIN EVENT C.” Other fields of math don’t care what happens when the stated conditions aren’t met, so this difference never comes up there. Probability does, but we misuse the word.

There is a lot of extra meaning in the fact that capitalized part. For example, like I said before, if “given” provides this meaning, then in the Monty Hall Problem, “given that Monty Hall opened Door #3 to reveal a goat” means that Monty opened Door #3 BECAUSE it had a goat, regardless of the other unchosen door. This makes the probability that the contestant’s door already has the prize 1/2, and I doubt Dr. Devlin would agree that “given” implies that.