Re: copying related probability question

Hal Finney (
Sun, 21 Sep 1997 10:57:07 -0700

I realized after I posted that long message that I had made a mistake
after all. Goes to show what happens when I stay up past my bedtime.

The problem is that it is wrong to neglect rare sequences of coin flips.
A series of 6 heads in a row has a probability of only 1/64, but OTOH
it produces 64 people who have all seen 6 heads, which largely compensates
for the low probability.

Suppose we ran the experiment I showed 64 times in parallel, once with
each of the 64 possible sequences of heads and tails. We'd start off with
64 people, and we'd end up with a lot more because of all the cloning (729
people, as it turns out). One group would see a series of all heads,
and there would be 64 of them at the end. One would see all tails,
and there would be only 1 of them at the end. We can chart it like this:

Number of Number of Number of people Number of
heads seen groups per group at end people total
6 1 64 64
5 6 32 192
4 15 16 240
3 20 8 160
2 15 4 60
1 6 2 12
0 1 1 1

So there will be 64 people who saw 6 heads, 192 people who saw 5, 240
people who saw 4, etc. From this we can calculate the average number
of heads that was seen by a participant who leaves the experiment, and
it is exactly 4, so the average fraction of heads seen is 4/6 or 2/3,
in accordance with Wei's option (B).

However, I am not 100% sure that this definition of probability is the
right one to use. Consider, for example, if someone offered to bet you,
before you enter the experiment, about the outcome of the coin flip.
Suppose you were willing to give him 2-1 odds that it would come out heads
(because you believe there is a 2/3 chance of that.) You put up $200
and he puts up $100.

After the experiment, if it came out tails, you lose your $200, but there
is only one of you. If it came up heads, you win $100, but there are two
of you, so you have to split it and each gets $50. Subjectively, there is
a 1/3 chance of losing $200 and a 2/3 chance of winning $50, which is a
losing proposition.

You need to make the bet at even odds in order not to lose money. In
that case there is a 1/3 chance of losing $100 and a 2/3 chance of winning
$50, which works out evenly. So despite the fact that subjectively you
will find at the end a 2/3 chance of heads, operationally you must base
your decisions on the assumption of a 1/2 chance of heads. The same cloning
which increases the numbers of copies for certain outcomes also divides
any returns which may come from those outcomes.

So it may be that the most successful people will be those who adopt a
definition of probability which does not take cloning into effect, and
therefore that is the definition which will win.