Approximating a probability from independent trials

Question

I am running simulations for a particular problem, which is not relevant to my question. Every simulation takes non-trivial amount of time and its result is binary - either 0 or 1, with probabilities $1-p$ and $p$ respectively. If I run $N$ simulations and get $x$ $1$'s, then $\frac{x}{N}$ is my approximation of $p$. My question is - which theorem/inequality is most commonly used to decide how big $N$ should be to be quite certain my error isn't too big? Does the answer change if I add additional information about $p$ (for example $p > 0.01$ and $p < 0.99$)? Obviously I would like $N$ not to be too big, to minimise running time of my simulations while still being quite certain that my approximation is accurate enough.

Answer 1

The standard deviation after $N$ trials is$\sqrt{\frac{p(1-p)}{N}}$. Since you have only an estimate for $p$, use $N-1$ rather than $N$ for estimating standard deviation. You need to decide how small the standard deviation has to be to satisfy your requirement.

The formula is from Bernoulli trials.