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Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn$i$ with probability $p_i$. Let $S$ be the resulting number of singletons,i.e., urns containing just one ball. We give an error bound for the Kolmogorovdistance from $S$ to the normal, and estimates on its variance. These show thatif $n$, $m$ and $(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i =O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$ tends toinfinity, and demonstrate an optimal rate of convergence in the CLT in thiscase. In the uniform case $(p_i \equiv m^{-1}) with $m$ and $n$ growingproportionately, we provide bounds with better asymptotic constants. The proofof the error bounds are based on Stein's method via size-biased couplings.

Normal Approximation for Isolated Balls in an Urn Allocation Model