Note that P(θ) does not necessarily represent the probability of a positive response from a particular individual. It is certainly feasible that an examinee might definitely select an incorrect answer, or that an examinee might know an answer for sure, based on the prior experiences and knowledge of the examinee, apart from the trait level. It is more correct to think of P(θ) as the probability of response for a set of individuals with ability level θ. Said another way, if a large group of individuals with equal trait levels answered the item, P(θ) predicts the proportion that would answer the item correctly. This implies that IRT models are item-invariant; theoretically, they would have the same parameters regardless of the group tested.
An assumption of these IRT models is that the underlying trait is unidimensional. That is to say, there is a single underlying trait that the questions measure that can be theoretically measured on a continuum. This continuum is the horizontal axis in the plots of the curves. If there are several traits being measured, each of which have complex interactions with each other, then these unidimensional models are not appropriate.