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Prob. that the spin is \(\uparrow\) is
\[ |\langle \uparrow, v \rangle|^2 = |\langle (1,0), (v_1, v_2) \rangle|^2 \]
\[ = |v_1|^2 \]
Prob. that the spin is \(\downarrow\) is
\[ |\langle \downarrow, v \rangle|^2 = |\langle (0,1), (v_1, v_2) \rangle|^2 \]
\[ = |v_2|^2 \]
