When sampling a population for surveys we can often limit our sample size to hundreds, but when doing a Monte Carlo simulation we need way more. Why? I’m a bit of a stats-noob, so I am not sure I will manage to formulate this question properly, but let me do my best.
I‘m trying to develop an intuition for sample sizes and when they are sufficient to get to a reasonable degree of representativeness of a larger population.
I know that when doing surveys or polls, a sample size of mere hundreds or thousands is often sufficient, even for very large populations. (This is at least what calculators such as this one are telling me.)
So now what happens if we extend this notion to Monte Carlo simulations of a multi-parameter model? Every simulation can be thought of as representing a sample of the larger population of all possible simulations (the solution space).
The solution space of a multi-dimensional problem can be very large, but the survey/poll scenario tells me that this doesn’t mean that the sample size will need to be large.
This seems to suggest that any application of the Monte Carlo method can be concluded within several hundred/thousand of simulations as well. But I feel this cannot be true, as it would appear to defeat the purpose of the plethora of “more efficient alternatives” to Monte Carlo (such as Markov Chain Monte Carlo).
What am I missing? What is the difference between surveying a large population of people on their election vote (for example) and “surveying” a large population of parameter permutations on their performance (when plugged into a certain model)?
Why does a surprisingly small number of surveys suffice to get a decent idea of the likely outcome of an election, while a small number of Monte Carlo simulations does not suffice (I think) to get a decent idea of the likely (top) performance of a model?
My apologies for needing too many words, but it’s the best I can do I am afraid. I hope you catch my drift regardless and can steer me in the right direction. Thanks a lot!

Population size is misleading. Consider the following two experiments:
Experiment 1: A coin lands on heads with unknown probability $p$. We flip the coin $100$ times and try to estimate $p$.
Experiment 2: We sample uniformly from a bag containing a trillion balls, either red or blue. The probability of sampling a red ball is $p$. We sample $100$ balls and try to estimate $p$.
These experiments might seem very different, but probabilistically they are extremely similar. Abstracting away the population size, we are trying to compute a single parameter value, $p$, and the value of $p$ has such a big effect on our observations that even with only $100$ samples the value of $p$ can be well-estimated. With more complicated models, there may be many parameters whose impact (individually and collectively) on our observations is very subtle, and so many samples are needed to tease out these subtleties.
This line of thinking leads one into the intersection of probability theory and information theory. Essentially, a few observations of flipping a coin carry a lot of information about the parameter $p$, but for more complicated models the amount of information in each individual observation may be very small. It is this that governs how many samples you need, and not the total size of the available data.
