A spin-$1/2$ particle is in the state $$|\phi\rangle=\frac13\begin{pmatrix}2+i\\2\end{pmatrix}.$$ The probability that measurement gives $S_x=+\hbar/2$ is
The state is normalized because $(|2+i|^2+|2|^2)/9=(5+4)/9=1$. The positive $S_x$ eigenket is $(1,1)^T/\sqrt2$. Its amplitude in the supplied state is $(2+i+2)/(3\sqrt2)=(4+i)/(3\sqrt2)$. The probability is the squared modulus, $|4+i|^2/18=(16+1)/18=17/18$, giving D.
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