“The probability of obtaining $+ \frac{\hbar}{2} $ in a measurement of the observable $S_{z}$ on a system in the state $\vert \psi \rangle $ is $\mathcal{P}_{+}=\left\vert \langle +\vert \psi \rangle \right\vert^{2}$.”
This is also a good time to analyze the units of $\hbar$. Start by asking what the units are, and lead the students to the fact that $\hbar$ has units of angular momentum. So, the observable $S_{z}$ makes a measurement on some kind of angular momentum-unit property of the particle, called the intrinsic angular momentum.
Also mention that in the postulate, the plus signs can be replaced by minus signs to represent the same postulate for the spin down case. Rather than writing the postulate twice, however, the plus sign can just be replaced by $\pm$ to represent both cases in one postulate and the sign can be chosen at will.