This question may turn out to be trivial, but it's worth asking. I'm working through Griffith's Introduction to Quantum Mechanics, problem 4.32 part a.
a) If you measured the component of spin angular momentum along the $x$-direction, at time $t$, what is the probability that you would get $+\hbar/2$?
My linear algebra is pretty weak, which may correspond to my problems in understanding the material. All the sources I've found online about probability in a specific direction say "the probability is just $|\langle \psi_x|\psi\rangle|^2$", and don't elaborate on why this is the case.
I understand that the probability is simply the absolute value of the wave function squared, but I'm not sure why, when seeking the probability in a certain direction, you take the inner product of the component in question and the wave function. i.e., $\langle\psi_x|\psi\rangle$.
Could someone point me in the direction of a proof regarding this, or give me a general idea of why this works?
Thank you.