= Solution
For the <Gibbs entropy>
$$
S=-k_B\sum_n p_n\log p_n,
$$
first maximize subject only to $\sum_np_n=1$ over $\Omega$ accessible microstates. A <Lagrange multiplier> gives
$$
-(\log p_n+1)+\alpha=0,
$$
so every $p_n$ is equal. Normalization yields the <microcanonical ensemble>
$$
\boxed{p_n=\frac1\Omega}.
$$
For the canonical ensemble, impose both normalization and the mean-energy constraint $\sum_np_nE_n=E$. Stationarity of
$$
-\sum_np_n\log p_n-\alpha\sum_np_n-\beta\sum_np_nE_n
$$
gives $p_n=C e^{-\beta E_n}$. The <canonical partition function> fixes $C$, so
$$
\boxed{p_n=\frac{e^{-\beta E_n}}{Z}},
\qquad
Z=\sum_ne^{-\beta E_n}.
$$
Thermodynamic consistency identifies $\beta=(k_BT)^{-1}$. This is the <maximum-entropy derivation of equilibrium ensembles>.
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