Theoretically, if I am rolling two fair 6 sided die and I want to minimize the number of times I have to roll them... for any outcome/payoff I want, how would I minimize the objective function of min(rolls)?
For instance, if I wish to receive payoff for rolling an 8, how would I state a mathematical expression for min(rolls) for a Payoff = 8?
First, given that we roll two dice, the probability that we get an $8$ is $\frac{5}{36}$. A simple counting argument shows this (the only pairs of dice that add to $8$ are $(2,6),(3,5),(4,4),(5,3),(6,2)$. There are $5$ of these. The total number of combinations of dice rolls is $6\times 6 = 36$. So, $5/36$). So, let $p = 5/36$.
Now, the Geometric Distribution gives the probability that you need $k$ bernoulli trials to get 1 success. Rolling an 8 with two dice is Bernoulli with $p = 5/36$, so if we let $X\sim\text{Geom}(5/36)$, we just want to find $E[X]$. The mean of the Geometric is: $$E[X] = \frac{1}{p} = \frac{36}{5} = 7.2$$
To find this generically, first find the probability of rolling whatever value with $2$ dice (these are easy to find), and then the expected number of rolls to get that number will be $1/p$.