= Solution
For an axis halfway from $+z$ toward $+x$, let $|n+\rangle=\cos(\pi/8)|\uparrow\rangle+\sin(\pi/8)|\downarrow\rangle$. The <Born rule> gives
$$
\boxed{p(n+|\psi_+)=\frac{1+1/\sqrt2}{2}\simeq0.854,\qquad
p(n+|\psi_-)=\frac{1-1/\sqrt2}{2}\simeq0.146.}
$$
The negative-outcome <probabilities> are complementary. Both outcomes can occur for both inputs, so one result cannot distinguish them perfectly. Repeated measurements on independently prepared copies distinguish their different statistics with arbitrarily high confidence. Reversing the diagonal direction exchanges the two numbers. This separates single-copy certainty from statistical distinguishability.
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