Quine's account of probability is motivated by the same concerns as his account of modality, distrust of intensional notions and metaphysical largesse. Hintikka characterized Quine's distrust of modality as "one-world view", and argued that it is contrary to his own maxim of accepting what is indispensable in science.
Quine does offer a stripped down version of possible worlds in Propositional Objects. His "possible worlds" are mathematically described states of affairs material distributions in a space‐time that obey (known) physical laws. This is reminiscent of Carnap's "state descriptions" in The Logical Structure of the World (Aufbau), and makes clear that possible worlds are nominalist fictions. Individuals in them are "points in space", not rigidly designated invariants, a la Kripke. As Quine writes, "one thing good about this version of possible worlds, nevertheless, is that it stays within a clear extensional ontology".
Carnap used his "state descriptions" to develop a probability calculus, a purely analytic one, of course, just as necessity for Carnap reduces to analyticity. It is somewhat similar for Quine, probabilities are de dicto, subjective degrees of belief, and probability attaches to sentences, not events. He sees it as a quantitative version of propositional attitudes featured in modal logics, with similarly "common clay" content of limited utility. The same quotational device can be used to paraphrase probabilities out, e.g. "the probability of the sentence ‘It will rain tomorrow’ is 30%". In From Stimulus to Science we read:
"The central concern of statistical theory is probability and subjective probability is degree of belief. The recent study of subjective probability by Brian Skyrms and Karel Lambert has an explicitly epistemological orientation. [...] Subjective probability is degree of belief. It dominates the normative side of naturalized epistemology, as noted at the end of chapter IV. It is a quantitative refinement of a propositional attitude and admits a formulation de dicto with the help of quotation in the manner of other propositional attitudes."
This would not, of course, work for quantum mechanics where probabilities are objective. Quine does not really elaborate much on it, but in Confessions of a Confirmed Extensionalist he seems to favor the same approach as Popper, based on propensities:
"Probability? Of events? Restore extensionality as above. Of states of affairs? Treat quotationally, like de dicto. How much remains e.g. in quantum mechanics? Maybe (as Føllesdal suggested) must settle for propensities, but still extensionally."
Propensity is a long term tendency to yield outcomes with stable frequencies. Since the underlying nature of this tendency is unclear propensities are often criticized for being mystical, for single events they are also untestable. Frequentist accounts seem to be more "extensional", but they do not seem to apply to single events. For the relationship between propensities and frequentism see Does the propensity interpretation of probability rely on the principle of indifference? Presumably, Quine should favor statistical interpretations of quantum mechanics that dismiss ontology and declare questions about transition probabilities to be the only legitimate questions. A critical survey of Quine's position is Chatti's Extensionalism and Scientific Theory in Quine’s Philosophy:
"The mathematical character of physics makes it also more extensional than other fields. But problems arise when physical objects are not identifiable, as is the case with quantum particles and when the methods used are not entirely deductive. Therefore, physical theories do not admit the same degree of extensionality, depending on the objects they consider and the methods they use, which shows that what seems criticizable in Quine’s approach is its radicality, which does not take into account the different approaches and methods used in natural science."