= Solution
By the <Born rule>, the <probability> for a measured <energy eigenvalue> is the squared <norm> of the <orthogonal projection> onto its whole <energy eigenspace>. Thus
$$
\boxed{\Pr(E=0)=\frac23,\qquad
\Pr(E=E_\star)=\frac16+\frac16=\frac13.}
$$
Every other <energy eigenvalue> has <probability> zero. These <probabilities> are independent of $T$: <unitary time evolution> under the unchanged <Hamiltonian operator> changes the <dynamical phases> but not the squared expansion coefficients. The two equal-energy terms describe one measurement outcome.
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