= Solution
The <Born rule> uses the squared modulus, so the imaginary term contributes positively: $|a+ix|^2=a^2+x^2$. For $a>0$, <wavefunction normalization> would multiply the supplied expression by $N=\sqrt{3/(4a^3)}$. Equivalently, take the ratio of the integrals directly:
$$
P=\frac{\int_{a/2}^a(a^2+x^2)\,dx}{\int_0^a(a^2+x^2)\,dx}
=\frac{19a^3/24}{4a^3/3}
=\boxed{\frac{19}{32}}.
$$
Using $\psi^2$ instead of $|\psi|^2$ would fail to produce a real, nonnegative <probability density>.
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