As I understand it, a pure quantum state is one that can be represented as a ket $\lvert\psi\rangle$ in a Hilbert space, and it contains all the information about the state of the system. As such, we have complete information about the state of the system. Why then is the information we extract from the state $\lvert\psi\rangle$ still probabilistic? That is, why are the observables that we measure still associated with a probability? Is it simply that although we have complete information about the state of the system, by "complete information" it is meant that we know all the possible values that each given observable of the state can take?
One reason I ask, is that in the case of an open quantum system, we have to use mixed states in which we have incomplete information about the state of the system. This involves a classical probability since it is the lack of information that causes the probabilistic description of the state of the system, however there is still a "quantum probability" associated with such a system, like in the case of a pure state, since its observables have an intrinsic probabilistic nature.
I find it confusing having the two, how to distinguish between them and why this so-called "quantum probability" arises, particularly when in the case of a closed or isolated quantum system we can use pure states which supposedly contain complete information about the observables of the system?!