Message-Id: <v01540b00b34d76fe4247@[209.204.150.77]>
Date: Wed, 28 Apr 1999 18:44:19 -0800
To: "Pinhole Listserv" <pinhole@exploratorium.edu>
From: cwings@buzz.sonic.net
Subject: Re: probability
Finally, a pinhole subject for the mathematicians among us! I've been
posing this problem to friends and relatives (my intuition says 50-50 in
spite of the coin analogy), and got this response from my brother, which I
thought was worth sharing:
OK, this thing has been bugging me. I think this is more complicated
than we've been led to believe.
It seems to me that there are two different problems here, that
consist of counting two different (but related) sample spaces. One is
the space of "families with two children, at least one of whom is a
girl," and the other is the space of "girls with one sibling."
If you count the families space, then indeed the number of families
with a boy and a girl is twice as big as the number of families with two
girls. And if you can show that the older child is a girl (or the
younger one, for that matter), then you have eliminated one quadrant and
the numbers even out.
But if you count the space of "girls with one sibling," you get a very
different answer. You get exactly half of them having a brother, just
as your intuition would expect.
To see this, consider a universe of 3 families: one with an older
brother and younger sister, one with an older sister and younger
brother, and one with two sisters. If you count families, you get 2
with brothers versus 1 without. But if you count girls, there are 4 of
them, 2 of whom have brothers and 2 of whom do not. The numbers are
different because you count both girls from the family with 2 girls.
The question now is which of these 2 counting schemes applies to the
given problem. As you described it to me, the problem was: "You have a
female friend who has one sibling. What is the probability that her
sibling is male?" It seems pretty clear to me that you are counting
girls here, rather than families.
On the other hand, if the problem was "You have a friend who has 2
children, at least one of which is a girl. What is the probability that
the other child is a boy?", then I think you are counting families. But
of course you can recast this question to be very similar to the other
one, like "What is the probability that that girl's sibling is a boy?"
The real lesson of all this, I would say, is that word problems can be
very subtle. The thing that really irks me about Marilyn Vos Savant
(who I think popularized this problem) is that she only talks about one
interpretation of her problems, which she loudly proclaims to be the
correct interpretation. In this case, it sounds to me like she latched
onto the wrong interpretation.