Recent results have estimated the error when sums of dependent nonnegative integer-valued random variables are approximated in distribution by a Poisson variable. Two problems are considered where these results can be used to provide simple solutions. The first problem studies the asymptotic behavior, as $\alpha \rightarrow 0$, of the number of independent random arcs of length $\alpha$ needed to cover a circle of unit circumference at least $m$ times $(m \geqq 1)$. The second problem deals with urn schemes.
"Two Applications of a Poisson Approximation for Dependent Events." Ann. Probab. 5 (5) 787 - 794, October, 1977. https://doi.org/10.1214/aop/1176995720