The details of the numbers below are not important, so I will just pick
some to give an example.
Ten people are selected at random and brought into a room. There,
two dice are fairly rolled. If they roll double sixes, which should
happen one time in 36, the people in the room get a reward and the
experiment ends.
If some other number is rolled, the people leave the room and the process
is repeated, except this time 10 times as many people are brought in.
We brought in 10 people the first time, so now we bring in 100 people.
We repeat the dice roll. If we get double six, we give out the reward
and the experiment ends; otherwise we choose a new group of people,
10 times more numerous, or 1000 people in this case.
We keep this up until we give out the reward, choosing 10 times more
people each time we fail. (Assume the population from which to choose
is infinite so we will never run out of people.) Then the experiment
is over.
Now, the question is, suppose that you find yourself selected as one of
the subjects in the experiment and you are brought into the room. You
watch as the dice are rolled. What should you expect to see? With
what probability will you expect the dice to come up double sixes?
By normal reasoning, you would expect the probability to be 1/36. The
dice are rolled fairly and nothing should diverge them from this result.
However there is another argument which suggests that the chances are
much higher. The idea is that approximately 90% of the people who ever
go into that room are going to see the dice roll as double sixes. That
is because we keep increasing the number of people after each failure,
so that when we do get a success the number of people who see that will
considerably outnumber all of the people who saw failures.
Since 90% of the people who go into the room see double sixes, it follows
that if you are a person in the room, the chances that you will see double
sixes is 90%, far greater than the 1/36 chance we calculated before.
This represents a sort of reverse causality where events in the future
can in effect influence the probability of events in the past. It relates
to the issue of whether our existence here and now is influenced by the
size of the human race in the far future.
My personal opinion is that a random person in the room actually would
see the dice roll as sixes, and that if I found myself in the room I
would bet on the double sixes if someone offered me this option. What
do you think?
Hal