Fair warning: I am not a math expert (that's why I'm here).
I would like to be able to calculate the probability of rolling a certain side on a die with n sides where any number of those sides has an unfair advantage over the others.
For example: On a 6-sided die, we could assume that the 1 - 6 sides are longer than the other sides due to wear and tear (or someone shaved the 1 or 6) and, therefore, more likely to come face up on a roll. I know that, on a perfect die, the roll is uniformly random and each side has a 1/side chance of coming up.
How do I calculate the probability of each side if sides 1 and 6 are longer?
What if sides 1 and 6 are longest, 2 and 5 are second longest, and 3 and 4 are shortest?
What if I'm using a 12-sided die? 10-sided? 20-sided?
I'm looking for a formula that I can plug numbers into and I'd really, really like an explanation of how it all works, if possible.
I found this link that talks about using Mathematica to calculate one side shaved on a 6-sided die, but I don't know how this changes when you increase the number of dice, or the syntax being used. I feel like the equation below (from the link) makes sense somewhat (in that σ represents the increase in probability of a 1 or 6, but I would like to know how to calculate σ.
f = { 1/6 + σ, 1/6 - σ/6, 1/6 - σ/6, 1/6 - σ/6, 1/6 - σ/6, 1/6 + σ };
Could I use the same formula to represent a 20 sided die?
f = {1/20 + σ, 1/20 - σ/20, ... ... ... ..., 1/20 - σ/20, 1/20 + σ}
Note: I took an intro to Statistics in college, but that's all the exposure I've had. I would like to understand more, but this is where I am now. I appreciate any help you can give me and any resources you can point me to.