On Mon, 12 Jun 2006, James Propp wrote:
For the second question, let's assume (for definiteness) that one child in a thousand is named Bartholomew. So, of the two million families with one boy and one girl, there are 2000 families with a boy named Bartholomew, and of the one million families with two boys, there are 1000 two-boy families in which the older boy is named Bartholomew and 1000 two-boy famlies in which the younger boy is named Bartholomew. (There may also be 1 family in which both boys are named Bartholomew --- cf. the Dr. Seuss poem "Too Many Daves" --- but they are too few to skew the demographics by much.) That makes 2000 two-boy families in which at least one of the children is a boy named Bartholomew. So the answer to the second question is 2000 divided by 2000+2000, or 1/2.
Interesting! You've convinced me, if we stipulate that each boy has a uniform probability of being named Bart (so that two-boy families have twice the chance of having a Bart). If we instead assumed that each family had a uniform pribability of naming a child Bart, I think my original claim would be correct, that it would be no different from the original problem of simply knowing that at least one child was a boy. Or, I suppose my assumption in my earlier message also works, but is even less, er, probable: if you made me dictator-for-life, and I decreed that every firstborn child be named Bartholomew, then knowing that there was a Bart wouldn't change anything. Heh, I was going to make a silly comment about "but what are the probabilities if the boy's name is *Steve*?", but then I realized that we could make it into one of Joshua's real-world kinds of problems by asking about boys named Mohammed. -J Trab pu kcip. Trab pu kcip!