From: Hal Finney (hal@finney.org)
Date: Wed Apr 30 2003 - 13:59:10 MDT
Eliezer writes:
> Here's another trick question that only the true Bayesian will
> resolve. You meet a mathematician. "How many children do you have?"
> you ask. "Two," he replies, "and at least one of them is a boy."
> What is the probability that they are both boys?
>
> I would *not* answer 1/3.
Rafal replies:
> 1/2 ?
The answer depends very much on your prior beliefs, as it should in a
good Bayesian problem. What do you think the mathematician's motives
are in asking this question? What might he have asked differently in
other circumstancs? How equal-minded is he with respect to the sexes?
Depending on the answers to these questions, I could see the answer
plausibly ranging between 1/2 and 1/3.
The "classic" answer to this puzzle is 1/3. The reasoning is that if
all we knew was that he had two kids, they could equally be girl-girl,
girl-boy, boy-girl, or boy-boy. Once he tells you that at least one
is a boy, that elminates girl-girl, leaving the remaining three with
equal probability. Therefore the chance that he has boy-boy is 1/3.
The problem is that this reasoning assumes that he would pose the puzzle
in exactly these terms for any of the last three cases. In particular,
if he has a boy and a girl, you have to assume that he would definitely
have said, "...and at least one of them is a boy." In other words,
you have to assume that he is a male chauvinist who views boys as more
important than girls.
If on the other hand he is a sexual egalitarian, then if he had a boy and
a girl he might have said, "...and at least one of them is a girl." If we
assume that in those cases there was a 50-50 chance that he might have
said "boy" or "girl", that gives us stronger reason to believe in boy-boy.
Only with boy-boy would he have certainly have said "boy"; with boy-girl
or girl-boy there was only a 50% chance he would have used that word. This
reasoning leads to an overall chance for boy-boy of 1/2.
Or you could make other assumptions; maybe he is more likely
to pose the puzzle if he does have two boys, since he knows that
mathematically-trained acquaintances will be likely to guess wrong.
In that case the probability might be even greater than 1/2. It all
depends on your priors.
Hal
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