🚀 go-pugleaf

In article <1k3f3hINN29k@gap.caltech.edu> carl@SOL1.GPS.CALTECH.EDU writes:
>
>OK, maybe if we do this by enumeration, it'll convince people.
>
... explanation deleted for brevity ...
>
>Thus there's a 50% chance that the sibling is female.  This is *NOT* the same
>as the "I've got two children and at least one of them is a girl" question. 

Excellent explanation.  Ennumerating the possibilities is the clearest way to
analyze this problem.

But, I bet a lot of people are still asking, "*WHY* isn't it the same as the 'I
have two children and at least one of them is a girl' question?"  I think this
confusion is the fundamental problem, and the reason that people still cling to
the 1/3 : 2/3 answer.

So in the original problem ("You see one of two children - the one you see is
a girl.  What's the other one?"), let's create an analog by flipping coins.
The problem becomes: you see one tail; what's the other coin?

We flip two coins, and pick one at random.  If it's a head, we throw out that
flip, and try again.  If it's a tail, we show it to you.  Over all possible
flips, here are the coins you will see (in caps):  hT, Th, Tt, tT.

The ones that get thrown out are Hh, hH, Ht, and tH.

So, of the four times you *see* a Tail, the unseen coin is a head twice, and
twice it's a tail.  Thus, the unseen coin is 50% likely to be a Head or Tail.

--

The other problem ("at least one of my children is a girl") is analogous to the
following coin-flip scenario:  We flip two coins, and if one of them is a tail,
we tell you "there's at least one tail."  We have now treated the Tt and tT
outcomes as a single event before you even hear about the throw.

If we flip HH, HT, TH, TT, then we will throw out the HH flip, and report success
on the HT, TH and TT.  In this new 'universe' of three possible outcomes, two of
them have a 'H' as the unreported coin, and only one has a 'T'.  Hence, the unseen
coin is 2/3 likely to be a 'H', and only 1/3 likely to be a 'T'.

--

To sum up, when you see one child, you may assume that child is selected at
random, and therefore the gender of the other child may be considered an
independent variable.  But when your friend tells you 'at least one child is
a girl', then the selection was *not* done at random; the gender of both
children was considered before telling you anything, so you may not treat the
gender of the other child as independent.

Hope that's clear.

-- Mike Maciolek		It is NOT an optical illusion.
mikem@ll.mit.edu		It just looks like one.