= Solution
For a normalized <quantum state>, $|\alpha|^2+|\beta|^2=1$, and the <Born rule> gives $\boxed{p(z+)=|\alpha|^2,\ p(z-)=|\beta|^2}$. If the displayed coefficients are not normalized, divide both by $|\alpha|^2+|\beta|^2$. A <quantum superposition> is a coherent linear combination: relative phase affects other <spin> measurements, unlike an incoherent statistical mixture with the same $z$-measurement <probabilities>.
Every pure <qubit> <quantum state> is an <eigenstate> of <spin> along some axis. Its <Bloch vector> is
$$
\mathbf n=(2\operatorname{Re}(\alpha^*\beta),\ 2\operatorname{Im}(\alpha^*\beta),\ |\alpha|^2-|\beta|^2).
$$
For real normalized $\alpha,\beta$, choose $\theta$ modulo $2\pi$ with $\alpha=\cos(\theta/2)$ and $\beta=\sin(\theta/2)$, allowing an irrelevant global sign. Then
$$
\boxed{\mathbf n=(2\alpha\beta,0,\alpha^2-\beta^2),\qquad
S_{\mathbf n}|\psi\rangle=\frac{\hbar}{2}|\psi\rangle.}
$$
Equivalently, $S_{\mathbf n}=(\hbar/2)\begin{pmatrix}\alpha^2-\beta^2&2\alpha\beta\\2\alpha\beta&\beta^2-\alpha^2\end{pmatrix}$. A <Stern-Gerlach spin measurement> on this axis yields its positive result with certainty, so the state is not a nontrivial <quantum superposition> in that measurement basis.
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