Re: Fun With Bayes' Theorem

From: Lee Corbin (
Date: Sun May 13 2001 - 15:16:27 MDT

Mark Galecki writes

>Lee responds:
>>The flaw in your argument is that the mathematician is obligated to
>>entertain no hypotheses concerning the procedure by which the parent
>>determined the truth or falsity of the statement "one of my children
>>is a girl". The only thing that matters is whether the statement is
>>true. It's not the case that the father is saying "I have picked...".
>I agree of course that the mathematician is obligated to entertain no
>hypotheses in addition to what the father said. But you are now quoting
>only part of the father's statement. Let me quote the whole relevant
>statement that you wrote:
>and I responded
>> Note the pair of phrases: "one", and "the other".
>Lee, did you carefully read your original message and this response??
>Why do I have to keep repeating them?

I did carefully read your response. Please allow that I just didn't
think that you were right.

Here was how you originally answered:

> However, this is not what Lee is really asking. Note the pair of phrases:
> "one", and "the other". What the father is really asking is "I have picked
> one of my children (not necessarily at random). That child is a girl. What
> is the probability that the other child is a boy". To this question the
> answer is 1/2.

But aren't we talking about the entire set of parents who could
truthfully say all this? Isn't it true that of all those parents,
two out of three of them in reality have a boy too?

Of course, if he were to say "my oldest child is a girl...", or
"the one my mother likes best is a girl...", then we no longer
have distinguishable cases. (My apologies Mark: I'm sure that
you already know that; I'm speaking to make sure that I'm not
misunderstood, and because I think that it is interesting.)

It's possible that your "focusing" query for linguists does indeed
somehow (it's by no means clear to me) isolate one of the children.

The rest of this post addresses Mark's question for linguists,
and in my opinion, has little mathematical interest.

> take the statement:
> "I have two children, sir", he says, "and one of them is
> a girl. What is the probability that the other is a boy?"
> In this statement, does the person:
> 1. focus on one particular child and then on the
> other child? (in this case the answer is 1/2)
> 2. does not focus on any particular child at any
> time (in this case the answer is 2/3)

I agree that it's likely that the person speaking has one
of them in mind, but not necessarily! If you had two girls,
and someone asked you, "do you have a girl", you might say
yes immediately without thinking of either one. You would
then be capable of saying "one is a girl!", "one is a girl!",
each subsequent time that you were asked. So we can contrive
situations where someone might make the above statements
without thinking of a particular child.

[Yes, this is a linguistic, not a mathematical consideration.
In mathematics problems, as you know, it's not important to
consider what someone may or may not be mentally focusing on.
The hard facts are all that matter. There is a certain
literalness, a certain trickiness, that appalls many people.
But it's even worse in courtroom law, where very literal answers
to questions are supposed to be considered very abstractly.
"Mrs. Jones has seven cows and all but two died. How many
survived?" It's immaterial what anyone is thinking. Riddles
share this same property.]

> 3. it is ambiguous and cannot be inferred whether he does or
> does not focus (in this case the original question that Lee
> posed is ambigous).

Some may claim that, and I cannot state categorically that they
are wrong. If enough people (of the right sort) don't like the
problem, then it is ambiguous and is a bad problem. But I like
my problem, and I think that it's fun. :-))))


This archive was generated by hypermail 2b30 : Mon May 28 2001 - 10:00:05 MDT