  Simple Monte Carlo is a versatile computational method with a convergence
rate of $O(n^{-1/2})$. It can be used to estimate the means of random variables
whose distributions are unknown. Bernoulli random variables, $Y$, are widely
used to model success (failure) of complex systems. Here $Y=1$ denotes a
success (failure), and $p=\mathbb{E}(Y)$ denotes the probability of that
success (failure). Another application of Bernoulli random variables is
$Y=\mathbb{1}_{R}(\boldsymbol{X})$, where then $p$ is the probability of
$\boldsymbol{X}$ lying in the region $R$. This article explores how estimate
$p$ to a prescribed absolute error tolerance, $\varepsilon$, with a high level
of confidence, $1-\alpha$. The proposed algorithm automatically determines the
number of samples of $Y$ needed to reach the prescribed error tolerance with
the specified confidence level by using Hoeffding's inequality. The algorithm
described here has been implemented in MATLAB and is part of the Guaranteed
Automatic Integration Library (GAIL).
