For example, if I want to determine the probability of getting tails when tossing a coin. By Monte-Carlo method, I toss the coin 1000 times and got 600 tails. As I know the distribution of the result is binomial, I can be confident to say the probability of getting tails is somewhere very near 60%. If I toss the coin another 1000 times it's likely that I get roughly 600 tails again.
So what about the general case? If I go over some Monte-Carlo simulation over and over again, what distribution of results am I expected to see? A normal distribution? Or the answer depends on the Monte-Carlo simulation itself?
EDIT: on the second thought this is a stupid question because clearly the distribution depends on the nature of the original problem. So I'm going to change my question to: why can I be confident when using Monte-Carlo method, and how to quantify my confidence?