= Solution
An <energy measurement> selects the <ground state> with <probability> equal to the squared <inner product> with $\psi_1$. Using $N=\sqrt3/L^{3/2}$ and <integration by parts>,
$$
c_1=\int_0^L\psi_1(x)^*\Psi(x)\,dx=\frac{\sqrt6}{L^2}\int_0^Lx\sin\frac{\pi x}L\,dx
=\frac{\sqrt6}{\pi}.
$$
Thus
$$
\boxed{\Pr(E=E_1)=|c_1|^2=\frac6{\pi^2}\simeq0.6079.}
$$
The full <Fourier sine series> has $c_n=(-1)^{n+1}\sqrt6/(n\pi)$, so $\sum_n|c_n|^2=(6/\pi^2)\sum_n n^{-2}=1$, an independent <wavefunction normalization> check. However $E_n|c_n|^2=3\hbar^2/(mL^2)$ for every $n$, hence $\sum_nE_n|c_n|^2$ diverges. This explicitly verifies the infinite-<energy expectation> qualification in part (b), without invalidating the individual projection <probabilities>.
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