BACKGROUND: I am designing a roleplaying game ("RPG.") I have been researching, and know some basics of probability and the normal distribution, including the equation to generate a standard bell curve.
Most RPG's use "polyhedral dice" of different denominations --- commonly ones with 4, 6, 8, 10, 12 and 20 faces (usually numbered from 1 to the number of faces) --- in order to determine whether or not the character a player is "roleplaying" in the game succeeds or fails in an attempted task.
These polyhedral dice are often added together in varying denominations (e.g. summing three six-sided dice or two ten-sided dice + one eight-sided die), to give a particular range of values and a particular mean with a more-or-less bell curve-shaped distribution of probabilities. In RPG's a roll adding together 'X' dice rolls of a die with 'Y' faces is denoted "XdY" --- so a roll of two ten-sided dice + one eight-sided die would be written as "2d10 + 1d8."
Strategies which I know can help one obtain results with particular desired distributional characteristics, with the curve of results shifted entire &/or skewed or 'squeezed' up or down the x- &/or y-axes, include...
- changing the number &/or denomination of dice rolled. (E.g. 4d6 ---> 6d4)
- using mathematical operations - add / subtract / multiply / divide (etc.) (E.g. 5d8 ---> 5d8 - 3; 6d10 ---> 6d10 / 3)
- keeping and adding together only 'N' dice results from the total number of dice rolled (the "dice pool") : these may be the N lowest or highest dice results. This method can be used to raise or lower the mean or variance of a probability curve --- or indeed alter both in different combinations. (E.g. "Roll 8d12." ---> "Roll 8d12, then add together the results from 3 of the dice rolled."; "Roll 10d20." ---> "Roll 10d20, then add together the 3 highest results rolled.")
I particularly prefer modifying a probability curve by keeping and adding the N lowest or highest rolls from a dice pool : I'm still working on an issue when it comes to picking lowest rolls, but keeping highest rolls nicely skews a bell-like probability curve to the right of the x-axis, raising the mean but tightening (lowering) the variance, as I desire.
QUESTION: If I have a probability curve of results with a bell-like shape and particular minimum, maximum, mean and variance values which I want to generate, is there some formula / method / tool to figure out which particular combinations of dice and numbers will generate what I want (or workably close enough to) using e.g. strategies like those I've noted above? (I'm not interested in other, more complicated ways of altering dice results like "exploding" dice, or "keep all dice rolled except the lowest" unless they prove necessary.)
I wish to generate a bell or bell-like curve with results from 0 to 50 --- with (as usual in a proper bell-curve) a mean equal to its median and mode midway on the x-axis, but the same variance as the curve generated by "3d100 / 3."
I wish to generate a bell or bell-like curve with results from 0 to 150 --- with (as usual in a proper bell-curve) a mean equal to its median and mode midway on the x-axis, but the same variance as the curve generated by "3d100 / 3."
Let's say I am playing a character in an RPG and I roll "3d100 / 3", with the output representing the result of my attempt to perform a particular task by rolling a specified target number or higher.
Now sometimes this roll should be modified to reflect the fact that my character's task is being made easier or harder due to positive or negative environmental factors at the time.
I can represent the effects of positive factors by rolling instead 4d10, then keeping only the highest 3, and dividing whatever this comes to by 3; I might write this as "(the 3 highest dice of 4d10)/3." The curve describing the output here is nicely skewed right, raising the mean but lowering the variance - exactly what I want.
But how do I represent the effects of negative factors making the task at hand harder? Here I want to skew the curve of results left, lowering the mean but raising the variance. However if I try 'flipping' what I did with the right-skewed curve above by rolling 4d10, then keeping only the LOWEST 3, and dividing whatever this comes to by 3 - i.e. "(the 3 lowest dice of 4d10)/3" - the mean does naturally lower, but the variance naturally lowers too...and without a specific method or tool to generate dice curves with the values I have chosen for me, I have been unable to find a way of avoiding this inverse relationship between left-skewed curves' mean and variance.
I really appreciate your help.