From: Hal Finney (hal@rain.org)
Date: Fri Dec 13 1996 - 13:40:00 MST
Thanks to David Musick for taking the time to explain some details
about the statistics of the shooting room paradox. While I don't
agree 100% with everything he said, it did shed a lot of light on
how some things would work.
In particular, David has convinced me that if people are selected
for the room not at random, but in a fixed order, and if you know
your position in that ordering, then you will in fact have a 1/36
chance of seeing doubles when you go into the room.
However there is another interesting question to ask about what you will
see in the room, supposing you are chosen at random: how many people are
expected to be in there with you? I find the answer rather surprising:
infinity.
There should be, on average, an infinite number of people in the room
with you.
I think this result further establishes that the paradoxical aspect of
the shooting room relies on the presence of infinities. Once you have
infinite sets involved much of our intuition goes out the window.
Without this realization, I think you still have a paradox, as with the
following argument:
A) Everyone who goes into the room has an equal chance of seeing double
sixes rolled.
B) 90% of the people who go into the room see double sixes.
If you didn't have infinite sets, I think these two facts in and of
themselves will serve to prove that the probability of an individual
randomly chosen from those who go into the room seeing double sixes
is 90%. The probability of an event is defined to be the limit, as the
number of trials approaches infinity, of the fraction of trials in which
the event occurs. If we repeat our experiment a large number of times,
the fraction of the selected people for whom the event "sees double sixes"
occurs will approach 90%. It follows, by the definition of probability,
that the probability of a person randomly chosen for the experiment
seeing double sixes is 90%.
(David suggested that this reasoning was invalid because different people
would be chosen each time. However this is not a reason to reject this
argument. When we say that a radioactive substance is such that 10%
of its nuclei will decay in time T, then if we repeat the experiment
a different 10% will decay. Yet even though the specific nuclei were
different in each case, we would not hesitate to say that each nucleus
faces a 10% chance of decay in time T. The issue of whether the same
nuclei are involved each time is not relevant.)
However, with infinite numbers involved things are more complicated.
The conditional probability of A given B, written as p(A|B), can be
expressed as p(A&B)/p(B); the probability of both A and B occuring,
divided by the probability of B occuring. If A is "sees double sixes" and
B is "selected for the experiment", then p(A&B) is the chance that a given
person will be both selected for the experiment and see double sixes,
while p(B) is the chance that he will be selected for the experiment.
p(A|B) is what we are interested in, the chance that someone selected
for the experiment will see double sixes.
With infinite sets, though, p(B) is 0, and p(A&B) is 0. So p(A|B) is 0/0,
which is indeterminate. It is no wonder then that different reasoning
leads to different results. We really can't apply simple arguments like
these to infinite sets and expect reasonable answers.
Hal
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