🚀 go-pugleaf

My two cents:

1.  A mother has two kids.  She tells you, "My older one is a daughter."
    What are the odds that the younger one is also a daughter?

2.  A mother has two kids.  She tells you, "I have at least one daughter."
    (Who knows why mothers volunteer this sort of bizarrely quantified
    information to complete strangers!  Anyway...)  What are the odds that
    the other one is a daughter?

The answer to question 1 is, of course, 1/2.  The children are specified:
older vs younger.  The answer to question 2 is, on the other hand, "Bad
question!  No referent biscuit for you today!"  Why?  Which one is the
other child?  The mother tells you, "I have AT LEAST ONE daughter."  But
she doesn't tell you which one, so the sex of "the other one" is an
unaskable matter.  Question 2 ought to be asked as follows:

2.  (Anyway...)  What are the odds that both children are girls?

In this case, the answer is 1/3.

Now, the question under examination is:

3.  A mother has two kids.  Walking down the street one day, you see one
    of the children, who happens to be a girl.  What are the odds that the
    other child is also a girl?

Does this correspond to question 1, or to question 2?  The distinction is,
question 1 names the female child, question 2 names no child.  Put this way,
it's clear that question 3 DOES specify the child: namely, THE ONE YOU SAW.
Thus, the following table can be created.

    child you saw     child you didn't see
1.        M                     M
2.        M                     F
3.        F                     M
4.        F                     F

Assuming no quibbles about reality, all four possibilities are equally likely
a priori, but after you see the child, you know that 1 and 2 are impossible.
You have no way to distinguish between possibilities 3 and 4; hence, the
probability that both children are girls is 1/2.

Careful reasoning is therefore necessary to distinguish between:

4.  A mother has two kids.  A close family friend says, "She has at least
    one daughter."  What are the odds that both children are girls?

5.  A mother has two kids.  A distant friend says, "I think she has at least
    one daughter.  Anyway, I saw a girl who said that she was a daughter
    of hers."  Assuming that no one has any reason to lie, what are the
    odds that both children are girls?

An amusing consequence of reasoning incorrectly about question 3 comes about
as follows.  Suppose you reason that the only combination eliminated is MM,
and thus, the probability that the other child is a girl is 1/3.  Ten years
from now, a friend of yours, unbeknownst to you, runs into the same child,
who in the meantime has had a sex change, and is now male.  She reasons as
you did ten years before, and "deduces" that the probability that the other
child is a boy is 1/3 (since FF is eliminated).  In this way, the probability
that the other child (that neither of you have seen, by the way) is a girl
has changed from 1/3 to 2/3, without either of you knowing anything whatsoever
about this person!  The resolution to this paradox, of course, is to realize
that the sex of the other child is equally likely to be male or female.

I don't have any formal training in this matter, but what I have said sounds
reasonable to me.  If anyone who has formal training can correct me on this,
please send me mail.  (In other words, I don't want any people who just took
first year probability sending me flames.  I took it too...)

byron elbows
(mail to brian@cs.ucla.edu)