Re: The "two child problem"

[Leigh Palmer <palmer@sfu.ca>]

> A man and a woman each have two children.  The man's
> older child is a son and at least one of the woman's
> children is a son.  Is either more likely than the
> other to have two sons?

OK, I'll have to be the philosopher.

I ask the straightforward question: How can knowledge
of a fragmentary *a posteriori* nature affect the *a
priori* probability of either person having two sons?
The answer, I think, is that it can't. The composition
of the families was determined before the information
came to light; it is not an appropriate question for
resolution by the methods proposed here. It is very
like the question "Given the present state of our
knowledge of the universe, how likely is it that its
origin was in a hot big bang followed by an epoch of
inflation to t=10^-34 second?" I have my own answer to
that question which relies not at all on the theory of
probability.

This is, as posed, is a question of *a posteriori*
probability. As such the answer to the question must
be "No." It also makes no sense to ask "What is the
proability that his (her) other child is also a son?"
If one asks it in that way it is still *a posteriori*,
but the assumption can be made that in either case the
"other" child has equal likelihood of being either
male or female. That assumption is clearly incorrect;
the child, like Schrodinger's cat, has a perfectly
definite sex when the question is asked.

I can formulate the question better so that the two
answers will be different. "What fraction of two-
child families having the older child male have two
male children?" and "What fraction of two-child
families having at least one male child have two male
children?"

This Marilyn is not bright enough to form her questions
properly. This reinforces my prejudice that high IQ and
intelligence are correlated more loosely than many
would imagine. I take it as a point of honor that I
have never acceded to an invitation to speak to our
local Mensa club. Imagine a club the members of which
share the delusion that performance on an examination
means something sufficiently substantial that it should
form the basis for choosing colleagues for social
intercourse!

Leigh