Fixing the p-curve for the S3 system

I’ve been looking at the dice rolling for the S3 skills system and it turns out the p-curve might be broken. This is my attempt to fix the p-curve.

Warning: maths and numbers ahead.

I was thinking about writing a simplified version of the Mad Science setting for S3 as a python game and I got to thinking about the probability distribution of the CL and DR system.

What is CL/DR?

CL stands for Challenge Level. CL is the number of dice that must be at or above the DR for the roll to count as a win. DR is difficulty rating and it indicates which number you need to get.

In theory, the way things work is that you have a limit number of dice. Thus as CL rises the task trends towards impossible. Meanwhile, DR models how many moving parts you need to be aware of and how tricky the task is.

I started to wonder if at the extreme DR values if things were “unfair”. It turns out that was the least of my problems.

I stated in the core rules that CL/DR moved in steps. I gave instructions for adding DR and upping CL. So things might work like this CL1/DR5 + 3DR = CL2/DR3 (something like that).

Why CL/DR cannot work like that

I plotted CL/DR values against various dice counts. To simplify, I assumed only six-sided dice. In my spreadsheet, I had the dice counts in row 2 and the DR in column B (CL was column A). This was the formula I used for the CL1.

1-(((6-(7-$B4))/6)^E$2)

What I am actually working out is the probability of failure. In this case, rolling lower than the DR. Then I deduct that from one to get the probability of success.

I figured that as the CL rose above 1, I could reduce the exponent (number of dice) relative to the CL.

1-(((6-(7-$B9))/6)^(C$2+1-$A9))

When I charted 5d6, this is what I saw

Not good.

When I reordered the CL/DR to put the probability in order this is what I got:

The CL and DR values jump around in a non-intuitive and non-elegant way. If I want a sliding CL/DR scale than players are going to be tied to a very weird chart of numbers.

Is CL/DR broken or is my p-math wrong?

The big question now is: Have I cooked up a broken system with CL/DR or did I get the probability math wrong? Whatever it is that is wrong, how can I fix it?

Right now, I have no idea.