>Note that one need not bring MWI into this at all. The only big
>assumption is the existence of a copy machine. Instead of MWI, one
>can think of the identical experiment being carried out on an
>ensemble of, say, 100 hapless souls Albert, Bernard, Caroline, etc.
>At time t1, some number close to 50 will have seen heads. At time
>t2, there will be 150 people, 100 of whom remember seeing heads,
>and 50 of whom remember seeing tails. From a bird perspective,
>if you picked any person at random from this group, the chance that
>they'll have seen heads is 2/3.
ok, let's aside the question of how to duplicate exactly a human being :)
I think the right answer is p=1/2 because this value will be naturally
observed by any Jane who repeats the experiment : if you toss the coin N
times, whatever you do with the Janes, they all will have seen the same
sequence of heads and tails, which entails statistically as much H than T.
Subjectively, suppose you toss your coin without knowing that the
duplicating machine is working (the machine can create another "you" in the
Andromedae Galaxy, say). You will not notice any difference, i.e. you
always will see a sequence of 50% H and 50% T.
With the color cards, each Jane will measure subjectively a probability 1/2
of yellow, 1/4 of red (1/2 H *1/2 "being chosen as Jane 1") and 1/4 blue ,
so again p(H) = p(T)=1/2 with the conditional probability formula.
The probability 2/3 is indeed the chance of finding someone who saw H after
the first experiment from a bird perspective, because duplicating
introduces a bias. However by repeating the experiment a lot of times, you
will find only very few people that have seen more H than T (in fact
exactly the same proportion as if there were no duplication). The bias
disappears statistically because for anybody it is as probable to have a
positive bias (H) than a negative one (T). .