Re: Shooting room paradox (addendum)

Eliezer Yudkowsky (
Sat, 07 Dec 1996 18:59:34 -0600

> If I collect some people at random, and show 90% of them a red ball,
> while I show 10% of them a green ball, wouldn't you agree that if you
> are selected, your chances of seeing a red ball are 90%? Why is it
> different if I show 90% of the people I select a double-six dice roll,
> while I show 10% a different value?

Your chances of being in a particular group *aren't* *random*.

A group is 90% red and 10% green. I roll two dice; if they come up
double-sixes, you're red. Otherwise, you're green. Then I assign the
rest of the group at random in a post-hoc distribution. YOUR chance of
being red is 1 in 36.

Any particular GROUP has a 1 in 36 chance of being red. How can a group
with a 1 in 36 chance of redness be composed entirely of people with a
90% chance of redness?

(How can a post-hoc group with 90% redness be composed of people who had
a 1/36 *chance* of redness? Because the "chance of redness" does not
equal "redness". The situation is weird, but it is not impossible like
the one above. 90% of the people were there when a single 1/36 chance
happened to happen, thanks to the experimental setup.)

At the time of your dice-roll, you have not been selected randomly from
a post-hoc pool of subjects. 90% of the subjects *will* *be* red, and
you have a 1 in 36 chance of being red. If you are not red, the
experimenter sets up some post-hoc conditions - multiplying the number
of subjects by 10 - to ensure that 90% of the subjects will be red.

(Incidentally, I tried to present this paradox to certain relatives.
You probably wouldn't believe me if I had a transcript of what they
said. Highlights of the conversation:

"Did you say dice or ducks?"

"Do the ducks roll the dice?"

"Maybe you could coat the dice with something bad-tasting so the ducks
would spit it out like this: 'PTOO! PTOO!'"

"Wait! Ducks can't count!"

They aren't senile, just very silly. If you're wondering how I turned
out the way I did, let's just say my upbringing may have had something
to do with it.)

--       Eliezer S. Yudkowsky

Disclaimer:  Unless otherwise specified, I'm not telling you
everything I know.