Another example where simple probability assignment fails is from "Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements" by Haim Gaifman:( http://www.columbia.edu/~hg17/RLR.pdf )

A coin is tossed 50 times. Jack analyzes the sequence of 0’s (heads) and 1’s (tails) and concludes that the sequence is random, with nearly equal probabilities for each side. Then Jill informs him that the coin contains a tiny mechanism, controlled by a microchip, which shifts its gravity center so as to produce a pseudo random sequence–– one that passes the standard randomness tests but is in fact defined by a mathematical algorithm. [...] Jill gives Jack the algorithm and the outcomes of additional 50 tosses fully accord with the calculated predictions. Let h be the hypothesis that the outcome sequence coincides with the sequences produced by the algorithm. Let the evidence be e. Given e, it would be irrational of Jack not to accord h extremely high probability. Had P(h)–– Jack’s prior subjective probability for h––been non-zero, conditionalization on e could have pushed it up. But if P(h) = 0, then, P(h|e) = 0, for every e for which P(e) > 0. Now Jack’s prior probability accommodates the possibility of independent tosses with non-zero probabilities for ‘0’ and ‘1’. Therefore P( ) assigns each finite sequence a nonzero value, hence P(e) > 0. If P(h) = 0, then Jack cannot learn from e what he should, unless he changes probabilities in a way that violates the Bayesian prescription.

And it explicitly questions the modeling of 'unknown' possibilites as 0:

Jack has never considered the possibility of a coin that produces, deterministically, a mathematically defined sequence; hence, if the prior probability is to reflect anything like Jack’s psychological state, P(h) = 0. But could he, in principle, have made place for such possibilities in advance?