A method for simplifying manual random number generation? 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?

This is an interesting question from a simulation viewpoint as most simulation algorithms require the call to a uniform $\mathcal U(0,1)$ pseudo-random generator.
If $X_i$ is the number of 6's out of $n_i$ rolls,
$$X_i\sim\mathcal Bin(n_i,1/6)\,,$$that is,
$$\mathbb P(X_i=k) = {k \choose n_i} \dfrac{5^{n_i-k}}{6^{n_i}}\,.$$
Using a dice random generator over $\{1,\ldots,6\}$ rather than a Uniform generator to generate this distribution could proceed by


*the unbiased estimator method: Find a joint distribution on $(X,Y)$ such that there exists a transform $H(x,y)$ such that $H(x,Y)$ is an unbiased estimator of $\mathbb P(X_i=x)$ and accept draws with probability $H(X,Y)$. Drawback: it is unclear such an unbiased estimator is available at a cost lower than $\mathsf O(n_i)$. Or is it $\mathsf O(\log_6~n_i)$ ?


*the naïve method: Generate from the dice iid draws $Y_1,Y_2,\ldots,Y_{n_i}\in\{1,\ldots,6\}$ and count how many are equal to $6$. Drawback: this requires $n_i$ draws.


*the inversion method (p.85): Generate from the dice iid draws $Y_1,Y_2,\ldots\in\{1,\ldots,6\}$ until the cumulated probability of the outcome (using eg lexicographic order) fits between two consecutive values of the $\mathcal Bin(n_i,1/6)$ cdf. Drawback: this requires a random number of draws and is only approximate.


*the accept-reject method: Select a distribution $\mathbb Q$ on $\{0,1,\ldots,n_i\}$ with probabilities $p_0,p_1,\ldots,p_{n_i}$, which can be generated in $\mathsf O( \log_6(n_i) )$ rolls from a 6-face dice, compute the maximum
$$M = \max_k p_k \Big/ {k \choose n_i} \dfrac{5^{n_i-k}}{6^{n_i}}$$
and accept a draw $k$ from $\mathbb Q$ with probability
$$p_k \Big/ M {k \choose n_i} \dfrac{5^{n_i-k}}{6^{n_i}}$$
Drawback: the acceptance step implies further draws to mimick a Uniform draw and $M$ may prove quite large. For instance, using a uniform $\mathbb Q$ leads to
$$M = \max_k 1 \Big/ {k \choose n_i} 5^{n_i-k}$$


*the pseudo-uniform method: Generate from the dice iid draws $Y_1,Y_2,\ldots\in\{1,\ldots,6\}$, minus one, taken as digits of a uniform $\mathcal U(01,)$ variate (in base 5), until the number $R$ of digits (in base 5) is large enough for the intended precision, set
$$U=\sum_{i=1}^R 5^{-i-1}(Y_i-1)$$ and use a uniformly fast generator from Devroye, p.520 based on uniform generators. Drawback: this is not completely exact unless $R$ equates the real number representation on the local machine.
