Warning: This blog post is neither entertaining nor informative. I recommend that you just skip it.
Once again we are floating in a Newtonian universe described by its microstate S(t), but of course we are just ordinary observers who do not necessarily know it.
At t0 we perform a little experiment, the equivalent of tossing a coin, with two possible outcomes H(ead) or T(ail); At t0+D we know the result.
The omnipotent demon watches the outcome too and in the case of T reverts the Newtonian universe to
its previous state S(t0-D), which explains the title of this blog post; In the case of H she does nothing.
However, the omnipotent demon is not infinitely patient and therefore the universe would loop only N times through the same state(s), then it continues.
We know about all this, because the omnipotent demon was so nice to inform us about it in advance.
Now we try to calculate the probability for H and there are two different ways to do it:
i) We do not know the microstate and both H and T are equally likely for what we know about S(t0).
The fact that the demon will play some tricks at t0+D does not change anything, thus p(H) = 1/2.
ii) There are the following possible cases for this silly game: This is the 1st time we observe the experiment
and the outcome is H, which we denote as 1H. Then there is 1T and also 2T, 3T, ... NT.
Notice that there cannot be a 2H, 3H etc. because the universe is deterministic, if T was seen the first time
it must be seen the 2nd time etc.
Since we cannot distinguish between those cases, we have the probability p(H) = 1/(N+1).
Obviously, this has some similarity with the sleeping beauty problem. But I am not asking which of the two
probabilities is 'correct' and I am not even interested if this little thought experiment tells us anything
about a relationship between time and probability. I am asking a different question.
How do you understand the limit D -> 0 ?
Told you so...