In most languages, most of the syntactic dependency relations found in any given sentence are PROJECTIVE: the word-word dependencies in the sentence do not cross each other. Some syntactic dependency relations, however, are NON-PROJECTIVE: some of their word-word dependencies cross each other. Non-projective dependencies are both rarer and more computationally complex than projective dependencies; hence, it is of natural interest to investigate whether there are any processing costs specific to non-projective dependencies, and whether factors known to influence processing of projective dependencies also affect non-projective dependency processing. We report three self-paced reading studies, together with corpus and sentence completion studies, investigating the comprehension difficulty associated with the non-projective dependencies created by the extraposition of relative clauses in English. We find that extraposition over either verbs or prepositional phrases creates comprehension difficulty, and that this difficulty is consistent with probabilistic syntactic expectations estimated from corpora. Furthermore, we find that manipulating the expectation that a given noun will have a postmodifying relative clause can modulate and even neutralize the difficulty associated with extraposition. Our experiments rule out accounts based purely on derivational complexity and/or dependency locality in terms of linear positioning. Our results demonstrate that comprehenders maintain probabilistic syntactic expectations that persist beyond projective-dependency structures, and suggest that it may be possible to explain observed patterns of comprehension difficulty associated with extraposition entirely through probabilistic expectations.