= Solution
Inside the <infinite square well>, the stationary <Schrödinger equation> for unit mass is
$$
-\frac{\hbar^2}{2}\psi''=E\psi,
\qquad \psi(0)=\psi(\pi)=0.
$$
Its normalized eigenstates and energies are
$$
\boxed{\psi_n(x)=\sqrt{\frac2\pi}\sin(nx),
\qquad E_n=\frac{\hbar^2n^2}{2}},
\qquad n=1,2,\ldots.
$$
The general normalized time-dependent solution is
$$
\boxed{\Psi(x,t)=\sum_{n=1}^\infty c_n\psi_n(x)e^{-iE_nt/\hbar},
\qquad \sum_{n=1}^\infty|c_n|^2=1}.
$$
Immediately after removing the barrier, the normalized wavefunction is
$$
\Psi(x,0)=
\begin{cases}
\sqrt{4/\pi}\sin(2x),&0\leq x\leq\pi/2,\\
0,&\pi/2<x\leq\pi.
\end{cases}
$$
Its <Fourier sine series> coefficients in the full-well basis are
$$
c_n=\frac{\sqrt8}{\pi}\int_0^{\pi/2}\sin(nx)\sin(2x)\,dx.
$$
They are
$$
c_2=\frac1{\sqrt2},
\qquad
c_n=\frac{4\sqrt2}{\pi}\frac{\sin(n\pi/2)}{4-n^2}\quad(n\ne2).
$$
The <Born rule> therefore gives
$$
\boxed{
\mathbb P(E_n)=
\begin{cases}
\dfrac12,&n=2,\\[4pt]
\dfrac{32}{\pi^2(n^2-4)^2},&n\text{ odd},\\[4pt]
0,&n\text{ even and }n\ne2.
\end{cases}}
$$
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