# Estimating number of Monte Carlo runs for low-probability event

I would like to run a Monte Carlo simulation to identify the probability of an event E occurring. While these are Bernoulli trials, each run will incorporate random selection of several independent, uniformly distributed values. I expect the probability of E to be very low (p < .001, possibly on the order of p ≈ .00001).

Is there a simple way to estimate the number of runs I will need to say that, for instance, p < .001 with 99% confidence? I have seen methods for estimation of the number of runs, but they all caveat their poor fit for particularly small or large values of p.

As an aside, is the number of runs I need in any way contingent on the number of variables involved?

• You need stratified sampling for Monte Carlo Aug 2 '16 at 18:37
• Since you don't know $p$ in advance, you may want to use a sequential method such as inverse binomial sampling, which automatically adapts the number of runs to achieve a prescribed relative precision Aug 3 '16 at 11:52
• @Aksakal The variables involved are not at all suited to stratification, at least as I understand it. Aug 4 '16 at 14:42
• @Luis I think you are right about sequential methods. I was hoping there was some way I could easily estimate in advance what the number of iterations would have to be. Aug 4 '16 at 14:43
• The problem is, the required sample size would depend on the unknown p. That's when sequential methods come in handy Aug 4 '16 at 14:54