It is a very good question how probabilities would work when you start
thinking of cloning. We have had only a bit of discussion on this in
the past, sometimes in the context of the many words interpretation of
QM, sometimes in the context of cloning or mind cloning.
The way I look at it is this. Consider repetitions of the experiment,
and for each person who results, look at what fraction of outcomes he
has seen in his lifetime. The limit, as the repetitions approach
infinity, would be the probability.
This justifies probability on the basis of past experience. We expect
a coin to flip with 50-50 odds because we have always seen it work
that way in the past. I believe people whose predictions of probability
coincide with their later experiences of events will be most successful,
hence this definition of probability will spread and become accepted.
(However, there are some paradoxical aspects to this definition.)
In Wei's experiment, when we do the repetitions, each person who
results from each stage gets the experiment done to him. So after
each time the coin lands heads, twice as many people will be the
subjects of the experiment. Obviously we cannot keep this going for
too long!
I have to change the wording of Wei's questions slightly because now
the experiment is being repeated. Here is the wording I would use:
1. What is the probability that you were just cloned (that is, that you
were just created as a copy of another person on this last coin flip)?
2. What is the probability that the coin landed head up on the most
recent coin flip?
3. You were not cloned as a result of the most recent coin flip. What
is the probability that the coin landed head up?"
Here is a chart where I have tried to trace the people who result from
a sequence of coin flips. I show 6 coin flips, which for specificity
fall H,T,H,T,H,T. I think my results will be valid for other
plausible coin flips. We start with person 1, and after the first coin
comes up H, we have his copy, person 2. These people then participate
in further runs of the experiment. After 6 coin flips we have heads
3 times, and so there are 2^3 or 8 people at the end.
I mark with a (*) each person who finds that he is is a newly created
copy after the most recent coin flip.
1
|
H
| \
1 2*
| |
T T
| |
1 2
| \
H H
| \ | \
1 3* 2 4*
| | | |
T T T T
| | | |
1 3 2 4
| \ \ \
H H H H
| \ | \ \ --\ \---\
1 5* 3 6* 2 7* 4 8*
| | | | | | | |
T T T T T T T T
| | | | | | | |
1 5 3 6 2 7 4 8
Now, using this chart we can answer the questions above by simple
counting. Below I show the values which will be calculated by each
person, with the person number in the left column followed by the odds
he calculates from his history line.
1. What is the probability that you were just cloned (that is, that you
were just created as a copy of another person on this last coin flip)?
(Count fraction of stars on each history line.)
1 - 0/6
2 - 1/6
3 - 1/6
4 - 2/6
5 - 1/6
6 - 2/6
7 - 2/6
8 - 3/6
Average odds are 1/4. This number should be the same after any number
of coin flips.
2. What is the probability that the coin landed head up on the most
recent coin flip?
1 - 3/6
2 - 3/6
3 - 3/6
4 - 3/6
5 - 3/6
6 - 3/6
7 - 3/6
8 - 3/6
Average odds are 1/2. This will obviously be independent of the number
of coin flips.
3. You were not cloned as a result of the most recent coin flip. What
is the probability that the coin landed head up?"
(Ignore the starred items and calculate fraction of heads)
1 - 3/6
2 - 2/5
3 - 2/5
4 - 1/4
5 - 2/5
6 - 1/4
7 - 1/4
8 - 0/3
Average odds are 49/160. I'm too tired to work out what this will
limit to as the number of coin flips increase, but it is not far from
Wei's suggestion of 1/3.
So, assuming I haven't made an arithmetic mistake (which is possible;
it took me about 4 tries so far), I come out pretty close to Wei's
answer of A. 1) 1/4, 2) 1/2, 3) 1/3. It looks to me like these are
the average fractions of the time a group of participants will have
encountered "yes" answers when they leave the experiment.
Hal