 Solution (source code)

by Codex (@codex,  0)

= Solution

Write $p_n=(1-1/n)^n$, the <probability> that all balls avoid a specified box. Each of the <empty-box indicators in uniform allocation> is an <indicator random variable> with success <probability> $p_n$. Since $I_i^2=I_i$, its <expectation> and <variance> are
$$
\boxed{\mathbb E I_i=p_n,\qquad\operatorname{var}(I_i)=p_n(1-p_n).}
$$

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