As e.g. Una Stojnic argues, a natural-language sentence like
If the marble is big, then it’s likely red.
is actually an anaphora for
It’s likely that if the marble is big, then it is red.
Or if you want me to quote Stojnic's exact words on this:
the anaphoricity of modals is captured
by requiring that the restriction on the domain of quantification be retrieved in a way
similar to how the antecedent of a pronoun is—either provided by the context, or explicitly by the prior discourse. The way anaphora is resolved, in both cases, is determined by
discourse structuring mechanisms, in particular, mechanisms of discourse coherence. [...]
[As to Yalcin's 2012 example,] the modal ‘likely’ in the consequent, which is
searching for the most prominent epistemically accessible possibility, selects this possibility as the restrictor for its domain of quantification. The consequent is thus understood as
further describing the possibility introduced in the antecedent, providing the intuitively
correct restricted reading—the marble is likely red, given that it is big.
And this analysis is actually repeated/reasserted in Stojnic's paper [I've changed the premise/clause indices to refer to them as given in the OP's question in the quote below]:
A familiar view is that modals are quantifiers over possible worlds, but just which
worlds depends on the context (Kratzer, 1977, 1981). We can exploit this to argue
that the problematic counterexample can be explained away by maintaining that the
modal ‘likely’ in (P1) contributes a different semantic content than the one in (P2), due
to contextual effects on the interpretation of the two occurrences of the modal; and
so, (P2) and the consequent of (1) fail to contradict each other. Accordingly, (P1,P2,C1) is
not really an instance of MT.
This strategy captures the intuition that the consequent of (P1) talks about a restricted
(conditional) probability, while (P2) talks about an unrestricted one. The challenge is to
explain exactly why and how the context secures different (and intuitively correct) interpretations for the two occurrences of the modal. To do so in a non-ad hoc way is notoriously difficult.
Properly interpreted, the natural language construct P1 is basically a statement about a conditional probability:
Pr(red|big) > some "likely" threshold.
As Sven Neth argues these kinds of "chancy" paradoxes generally rely on the fact that (knowing) a conditional probability (really) implies nothing about the unconditional probability.
E.g. in this 2012 Yalcin/example problem,
Pr(big)) can be vanishingly small if there are very few red balls aside from the
big subset, which is actually the case here as there are many more small & blue balls, rendering [the unconditional]
The actual premises in proper form are:
P1: Pr(red|big) > likely-threshold
P2: Pr(red) < likely-threshold
In proper probabilistic reasoning, you can only infer about probabilities,
so the conclusion has to translated too in that form
C1: Pr(big) = 0
For the sake of making this a simpler calculation, using equalities rather than inequalities for the probabilities as given the problem setup, those numbers for the premises are
Pr(red|big) = 0.75
Pr(red) = 0.4
These allow you to infer (using the conditional probability formula) that
Pr(red⋂big) = 0.3 (i.e. red and big) but you cannot infer anything whatsoever about
Pr(big) from those two premises alone. (Which is basically Neth's point about these kinds of paradoxes, generally.)
In general, "probabilistic logic" is a "work in progress", meaning various formalism have proposed, but none (of the modern ones) are as "dumb" as what Yalcin suggests in that example, as far as I know.
To give you a basic insight here as to why this is difficult, a basic ‘probable’
operator (which I'm gonna call is-likely) was e.g. suggested by Hamblin with the meaning of exceeding some set probability value (e.g. 0.5). Alas doing much inference that "logic" way (instead of calculating probabilities) doesn't work too well because that operator is not a normal modal operator, meaning that
P1: x is-likely
P2: y is-likely
does not imply that
C: (x and y) is-likely.
Since the fact that there's some anaphora in Yalcin's natural-language P1 ("If the marble is big, then it’s likely red") was challenged in the comments, to say a bit more on that: if look more carefully at the [formal] MT Yalcin sets up here, he makes it a simple propositional matter, but that alone is a bad formalization because the words "marble" and "it" refer to the same thing in the natural language expression, but there's no referent in common between "φ" and "probably ψ" as Yalcin formalized that natural language statement as (P1) "φ → probably ψ".
To push the analogy (without resorting to probabilities) of why that lack of any syntactic commonality between φ and ψ is a bad formalization (for this), you might as well write a simpler rhetorical style argument that somewhat captures that idea:
P1: If Biden is the president of the USA, then I'm Biden.
P2: I'm not Biden.
C1: [Therefore,] Biden is not the president of the USA.
(As yourself why P1 might "sounds alright" at some pub, but it's actually not properly formalized merely as some "φ → ψ" where φ and ψ are propositions, in the formal sense of that word.)
And to include the obvious here, if you change P1 to something lacking an overt implication symbol, MT is not directly applicable anymore. You need the whole FOL + ZFC + probability space axioms (which also needs some construction of real numbers)
to syntactically reason about equations with probabilities symbols like "Pr". Simpler "alternative" systems (like Hamblin's) turned out not to be very useful in terms of what you can prove in them, relative to "reality" (which really is just a word here for the pre-theoretic, frequentist intuition we might have about a problem setup like that in the first quote you gave.)