= Empty-box indicators in uniform allocation
{title2=$p=(1-1/m)^n,\quad q=(1-2/m)^n$}
Place $n\ge1$ balls independently and uniformly into $m\ge2$ boxes. If $I_i$ indicates that box $i$ is empty, then $\mathbb E I_i=p$, $\operatorname{var}(I_i)=p(1-p)$, and, for $i\ne j$,
$$
\operatorname{cov}(I_i,I_j)=q-p^2<0.
$$
Indeed all balls avoid one specified box with <probability> $p$ and two specified boxes with <probability> $q$. The <indicator random variable> identities give the <expectations>, <variances>, and <covariance>. Finally $1-2/m<(1-1/m)^2$ proves strict negativity. Thus the number $K$ of empty boxes has $\mathbb E K=mp$ and $\operatorname{var}(K)=mp(1-p)+m(m-1)(q-p^2)$ by the <variance of a sum>.
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