  We consider the estimation of rare-event probabilities using sample
proportions output by naive Monte Carlo or collected data. Unlike using
variance reduction techniques, this naive estimator does not have a priori
relative efficiency guarantee. On the other hand, due to the recent surge of
sophisticated rare-event problems arising in safety evaluations of intelligent
systems, efficiency-guaranteed variance reduction may face implementation
challenges which, coupled with the availability of computation or data
collection power, motivate the use of such a naive estimator. In this paper we
study the uncertainty quantification, namely the construction, coverage
validity and tightness of confidence intervals, for rare-event probabilities
using only sample proportions. In addition to the known normality, Wilson's and
exact intervals, we investigate and compare them with two new intervals derived
from Chernoff's inequality and the Berry-Esseen theorem. Moreover, we
generalize our results to the natural situation where sampling stops by
reaching a target number of rare-event hits. Our findings show that the
normality and Wilson's intervals are not always valid, but they are close to
the newly developed valid intervals in terms of half-width. In contrast, the
exact interval is conservative, but safely guarantees the attainment of the
nominal confidence level. Our new intervals, while being more conservative than
the exact interval, provide useful insights in understanding the tightness of
the considered intervals.
