I am playing at making my own table-top gaming system/rules and I wanted to have a better handle on how likely different dice combinations will give a higher result than one another. I know that a six sided die roll averages to 3.5, and an eight sided die roll averages 4.5, but I still don't quite have a grasp on just how likely it is an 8 sided die comes up with a higher result than a 6 sided one.
I would also like to know how adding integers to die results effects their comparative advantage as well, like how often would the sum of 3 six-sided dice with a 1 added to the final result give a higher outcome than just 3 six-sided dice?
Thanks in advanced for any advice, I'm just not really sure where to start with this, I focused mostly on algebra/calc/trig in school and never really did any probability/stat.
Best Answer
For your first question, six-sided die vs. eight-sided, make a $6$ by $8$ table, with values $1, 2, 3, 4, 5, 6$ in one direction and $1, 2, 3, 4, 5, 6, 7, 8$ in the other direction.
The resulting $48$ small squares in the table determine $48$ possible outcomes of the two dice. You can then see for how many squares does six-sided beat eight-sided; how many ties; and how many times eight-sided beats six-sided. Assuming the dice are fair, you can then divide by $48$ to get the desired probabilities.