@EmilioPisanty In the simple case where we want to find the probability of say measuring some eigenvalue $a$ corresponding to some operator
$\hat{A}$, we then project onto the eigenstate to get
$\langle \psi|
a
\rangle
\langle a|
\psi \rangle$ which is the probability amplitude of the measurement yielding $a$, so this follows from one of the postulates of QM. In this case I guess we want to find the probability of the measurmement $\hat{S}$ yielding $\frac{\hbar}{2}$ after a sequence of measurements. Hence I assume that we must adapt the simple case that I gave. Am I right?