Bertrand Russell says that if a theory predicts something very improbable and it is found to be true, the theory is validated in proportion to the improbability of the supporting evidence. From an information standpoint, extremely improbably events do carry much more information. So being able to believe the right improbable things potentially becomes a gateway to more actual information.

So I do or I don’t, and it makes no difference whatsoever which it is. It can only be one or the other, from which follows the probability of .5 for the answer — Mww

No, it doesn't. You are misusing probability.

If you insist, so be it.

What is the relationship between defined subjective beliefs and undefined or unknown physical probabilities?

Consider an urn containing N balls, each ball being one of K colours. Suppose we express our 'subjective indifference' to the colour c of a ball randomly drawn from the urn by saying that our 'prior belief' is P(c)=1/K for c=1..K. On the surface, this assignment might look objectionable on the grounds that it is the physicality of the urn we are actually interested in, which could contain any proportion of balls of each colour. Yet our supposedly 'subjective prior beliefs', are in fact equivalent to saying that the actual ball colour frequencies inside the urn are undefined in our problem; for notice that the colour frequencies of balls drawn from every possible urn containing N balls of K potential colours, is 1/K.

In other words, in this problem our expression of a 'subjective' and 'definite' prior probability is rational, but it is actually a misnomer, since it is a frequency summary of logical deductions over the set of possible world states defined by our problem,in which the actual world-state is left undefined.

In conclusion, whenever a subjective belief is expressed and explicitly quantified in a problem, it should be understood frequentially in terms of a set of logical deductions over a set of possible worlds.