Re: Shooting room paradox (addendum)

Hal Finney (hal@rain.org)
Thu, 5 Dec 1996 21:27:54 -0800

From: abbot@alecto.physics.uiuc.edu (Jacob Costello)
> Here's a situation analogous to the shooting room where the probabilities are
> (I hope) more clear. Suppose we have an atom with thirty six possible
> orientations, each with equal probability to be found (a made up quanton).
> >From these atoms we're going to build inverted two-dimensional ziggurats.
> Each layer will be contain a factor of ten times as many atoms as the one
> below it. (a factor of three in the figure below)
>
>
> xxxxxxxxx
> xxx
> x
>
> We'll build layer by layer, and measure the orientation of the atom on the far
> right end of the layer after finishing a layer. If we measure the 36th
> possible orientation for that atom, we'll stop building. Suppose we have an
> infinite supply of these x atoms to build with. Surely what orientation we
> measure has nothing to with how many atoms are left, or even the fact that
> we're building a funny looking object.

This doesn't quite work as an analogy, because in the shooting room,
all observers see the same result when we roll our dice. So for your
atom model to work, we have to assume that all the atoms in each row
have the same orientation. Maybe there is some quantum coupling between
them which forces them all into the same state.

So when we measure the 36th orientation and we stop, we know that all the
atoms in the top row have that orientation. It follows that 90% of all
the atoms we have produced will be of this type.

>From another message:
> I think all this means (according to classical probability) is that if
> this experiment (an entire sequence of getting people and rolling and gettting
> more people until the sixes are rolled being one experiment) is conducted an
> infinite number of times, then any person will be in the room at the point the
> sixes are rolled in 90% of the experiments. The underlying reason of course
> is that more you have a higher probability of being in a large group than
> a small group, a priori.
> This is no contradiction to the 1/36 probability. Saying that there
> is a 1/36 probability of seeing sixes if you are in presently in the room
> refers to a different set of experiments, namely gathering a bunch of people in
> a room and rolling the dice. In 1/36 of these (infinitely repeated)
> experiments double sixes will be rolled (how many people are in the room is
> irrelevant.) What has happened is we've gained more information. Before the
> experiment starts we don't know which group we're in, but once we actually do
> know the probability is 1/36.

I think you are saying here that you agree that in the shooting room
experiment as I described it, the observers would be justified in expecting
to see the double sixes with 90% probability. But if did another kind
of experiment where we just put some people in a room and rolled some
dice, in that case we'd be back to 1/36.

It still seems strange to me to imagine being one of the people in
the shooting room. I am about to watch someone roll a perfectly fair
pair of dice, yet I expect this totally unlikely outcome to occur with
90% probability! It seems almost magical, as though an unseen hand is
reaching out to manipulate the rolling dice and bring them up double
sixes.

Hal