 Empty-box indicators in uniform allocation

Place $n\ge 1$ balls independently and uniformly into $m\ge 2$ boxes. If $I_i$ indicates that box $i$ is empty, then $\mathbb{E}{I}_i=p$, $\mathrm{var}(I_i)=p(1-p)$, and, for $i\ne j$,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 $\mathrm{var}(K)=mp(1-p)+m(m-1)(q-p^2)$ by the variance of a sum.