Is there a good way to take a more complicated dice-rolling mechanism and create a simpler dice-rolling mechanism that gives roughly similar results?
Example 1: Say you have a game where both sides roll a number of 6-sided dice (each side rolling a possibly different number of dice) looking for 6s. Unfortunately, other rules of the game frequently cause the number of dice rolled to be excessive (say, 40-60 or 80). The assorted statistics of the resulting distribution is fairly easy to calculate, but is there a way to use that to create a simpler mechanism (using, say, not more than 10 dice) that produces approximately the same results?
Example 2: Say you have a game where $n$ 6-sided dice are rolled looking for values greater than $x$, to identify possible successes. These successes are then verified by rolling 1 die per possible success looking for a value greater than $y$. ($n, x,$ and $y$ are parameters of the mechanism.) Is there a way to create a mechanism that produces approximately the same results with only one step?