There are two sources of uncertainty in the setting you have described.
First, there is sampling uncertainty: whether your sample distribution (i.e. the distribution of 20K randomly drawn values) is similar to the population distribution (i.e. your "known" distribution). By your description, your population distribution $X$ is discrete, with $k$ values in the support of $X$. If $k = 100$ and you draw $N = 10$ sample values, then it is not very likely that your sample distribution will closely match your population distribution (assuming the Pr$(x_i) >> 0$ for all $i$ in $(1,k)$ ). If you draw $N = 1000$ samples, then your sample distribution will be much closer, and so on as $N \rightarrow \infty$, thanks to the Central Limit Theorem and Law of Large Numbers. With your sample size of $N = 20,000$, the deviation is going to be quite small, and therefore sampling uncertainty will be close to zero.
Second, there is model uncertainty, which is the uncertainty associated with your "known" distribution. For discrete distributions, this can take several forms. You are using a distribution with $k$ possible values in the support, but maybe $k$ is not the right number (i.e. doesn't match reality). Maybe it's more, maybe it's less. For now, I'm going to assume that $k$ is the right number of possible values for Pr($x$).
If so, your major source of model uncertainty will be your estimates of the discrete probability for each possible value of $x$. These can arise through inaccuracy (i.e you estimate Pr($x_i) = 0.2$ when really Pr($x_i) = 0.1$), imprecision (i.e. you estimate Pr($x_i) = 0.3$ when really Pr($x_i) = 0.33333333\dots$), ambiguity (i.e. you estimate Pr($x_i) = 0.2$ when really Pr($x_i) = 0.1$ or Pr($x_i) = 0.3$), or incompleteness (i.e. you don't have an estimate for Pr($x_i$) so you interpolate using Pr($x_{i-1}$) and Pr($x_{i+1}$).
Perhaps the simplest way for you to deal with model uncertainty is to generate three population distributions. The first step is to estimate an upper limit and lower limit for each value Pr($x_i$). These should be "90% confidence" intervals. It is possible to elicit such estimated limits from experts after you have put them through a calibration exercise (see the book How To Measure Anything by D. Hubbard).
The second step is to generate the three population distributions. The first distribution you already have: your "expected population distribution". The second would be "lowest mean population distribution", where the estimates for individual values Pr($x_i$) are chosen from the upper or lower limits such that the overall distribution has the lowest mean value. Effectively, you are adjusting the probability estimate to the upper limit for the lowest values of Pr($x_i$) and the reverse for the highest values of Pr($x_i$). Then you reverse this procedure to produce the third distribution: "highest mean population distribution".
Once you have these three population distributions, you then run your simulation three times, generating 20,000 samples from each of the three distributions. You can then perform any statistical analysis you want on the resulting sample distributions (i.e. comparing sample means, sample variance, etc.) to yield an upper limit, a lower limit, and expected value.
This last step is optional, because as I said above, with 20,000 samples, the sampling uncertainty will be very low -- close to zero -- so you can do the analysis on the three population distributions themselves.
The above approach is simplified, but I offer it to you because you say you have a limited knowledge of statistics. This approach assumes that the uncertainty in estimates of each Pr($x_i$) is fairly uniform over all $i$ in $(1,k)$. If, instead, nearly all of your uncertainty is concentrated in just a few $x_i$, then you'd need to follow a different procedure -- Monte Carlo simulation.
