The book "Introduction to quantum mechanics" by Griffiths starts by introducing the wave function. The squared of the integral of the wave function gives you the probability of measuring the position of a particle between a and b. Later it talks about eigenstates and eigenvalues.The eigenstate describes the determinate state of the system after a measurement of an observable with operator $Q.$ The eigenvalues of $Q$ is what we can measure from $Q,$ i.e. $L_zf = hm_lf$ where $L_z$ is the angular momentum operator, f is the eigenstates and hm is the eigenvalues. An arbitrary state is ie $ \psi = c_1f_1 + c_2f_2$ where $c_1^2,c_2^2$ is the probability of measuring the eigenvalues corresponding to the given eigenstates.
My question is the following: why is the square of the wave function in the beginning giving us the probability of measuring the particle between a and b? Is it just that the wave function is a eigenstate of the position operator? If we take the squared of the eigenstates we get from measuring $L_z$ will we get the probability of measuring a particle between a and b?I know that the wave function just describes a system's state like the eigenstate.