Re: Fun with Bayes' Theorem

From: Lee Corbin (
Date: Sat May 12 2001 - 11:20:06 MDT

I had posed the problem

>> 2. A little girl's father discovers that his wife is
>> pregnant again (but they don't know the sex of the
>> unborn child). The man decides to visit a mathematician.
>> "I have two children, sir", he says, "and one of them
>> is a girl. What is the probability that the other is
>> a boy?" What did the mathematician tell him?
>> ***Answers*** For problems 1, 2, and 3, the answer is 2/3.

and Eliezer responded

>I don't think your word problems provide the necessary
>priors. To estimate the probabilities, I would need to
>know the algorithms used by each of the people making
>statements, and that information is not provided.

You're expressing how a statistician would answer. The father
didn't take the problem to a statistician, or to any sensible
person. He asked the question of a mathematician who plays
by rules that I discoursed upon in my last post. Any math
textbook is riddled by problems that would appear equally
problematic from the viewpoint you are advocating.

Then Eliezer brings up what is apparently a distinct problem:

>The "mathematician's daughters" is a much worse instance of the problem.

Okay, so now the subsets go as follows (I'm still following the
procedures used by mathematicians): The set of all parents is
reduced all the way down to the set of mathematicians with two
children (at least) one of whom is a girl.

>If you know that the mathematician will say "At least one of my children
>is a girl" for BG, GB, and GG, but say "At least one of my children is a
>boy" only for BB, and the mathematician says "At least one of my children
>is a girl", then the probability that the other child is a boy is 2/3. If
>you know that the mathematician will say "At least one is a girl" for GG,
>"At least one is a boy" for BB, and pick a statement at random for BG and
>GB, then the probability that the other child is a boy is 1/2.

Maybe mathematicians would be inclined to mean something different than
ordinary people would with these statements, maybe not. But for the
purposes of a math problem, it's immaterial. As I said in my last post,
all that matters is whether the statement "I have two children, and one
of them is a girl" is true. In the math problem, we really shouldn't
make any further hypotheses.

>I've *always* felt that riddle had a problem; now, twelve years later, I
>finally get to articulate it.

Which riddle? The one I asked (about the man whose wife is pregnant
and who cunningly---or stupidly---goes to a mathematician to talk it
over and carefully neglects to mention certain information), or the
"mathematician's daughter" problem that you just posed?

>(In Lee Corbin's question as stated, I expect that the mathematician
>hypothesizes the father to be symmetrical, rather than hypothesizing
>that the father exhibits a feminine bias, and answers 1/2.)

As I say, the mathematician's ideal role is to avoid hypothesizing,
and I have no idea what "feminine bias" means here. Given only the
actual statement that the father made, the answer 1/2 is incorrect.


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