Hey there. A while ago I came across a paper that tried to do a statistical analysis of the dice, that included the math, methodology and nifty graphs and everything, but I can’t seem to find it anywhere. Anybody know what I’m talking about, and knows where it is? Any help would be appreciated.
No idea, but this is a dice odds calculator:
Right. But it only predicts success/failure or advantage/disadvantage, but not both, and this is important because these to axes are statistically tied together. I remember the results of the study, though. Staistically, you’re more often going to roll a success with disadvantage, or failure with advantage because of the pip distribution across all the dice. This smooths out with the more dice you chuck, of course, and if you think about it while looking at the dice it kind of makes sense. I just wanted to find this study so I could show it to other people to kinda prove I’m not talking out of my butt. 
1 Like
One way to prove it yourself is to actually just raw total up the number of symbols on the dice and divide by the number of dice sides.
So since there are 9 success on a Proficiency die, you generate 0.75 Success on average, and with 8 Advantage, 0.67. A Difficulty die is 0.5 Failure and 0.75 Threat, meaning that a Proficiency die vs. a Difficulty die will, on average, generate 0.25 Success and 0.08 Threat.
Right. But this is still treating as if the pips of the axes are independent of each other, which they’re not. See, the more success you roll, that takes away from the number of advantage pips that are available to you, and vice versa. Save, of course, for a few sides that have both. Because of this, you have the skew that I mentioned. Anecdotally, I’ve noticed this trend, and the paper I mentioned laid out the math and methodology other than just looking at the dice and saying, “Well, that looks right.” 
It’s not a big deal, really. Just a bug in my head that my neruodivergent brain latched onto one day and spent an entire evening trying to hunt this thing down.
Oh I totally get it, I love that sort of thing as well (and am often teased for calculating odds in my head). Last night, to help myself fall asleep, I was calculating the odds of multiple symbols from Boost dice and comparing that to the effect of the Clanker Killer talent.
That skew is a big factor, but in a big enough picture, the law of averages asserts itself and you see the pattern I described. If you look at the dice and strip out double-effects, looking only at how many sides have a given symbol, you get the following results:
- Ability: 50% Success, 50% Advantage (generates 0.625 and 0.625)
- Proficiency: 58.3% Success, 50% Advantage (generates 0.75 and 0.67)
- Boost: 33% Success, 50% Advantage (generates 0.33 and 0.67)
- Difficulty: 37.5% Failure, 62.5% Threat (generates .5 and .75)
- Challenge: 58.3% Failure, 50% Threat (generates 0.75 and 0.67)
- Setback: 33% Failure, 33% Threat (generates 0.33 and 0.33)
As you add more dice in an equal and linear fashion, Proficiency and Challenge offset each other, while Ability and Difficulty create a distinct separation, with the odds of high success/high threat growing. The higher the difficulty, the more weighted a positive dice pool needs to be to generate consistent Advantage results. However, Boost operates as a real wildcard. It is both more likely to generate Advantage than a Setback and has a higher Advantage ceiling, conveying a direct and distinct advantage to the positive die pool at a rate of 16.7% Advantage generation and +0.33 Advantage per Boost/Setback pairing.
Ability/Difficulty meanwhile is a rate of 12.5% Success generation and +0.125 Success against 12.5% Threat generation and +0.125 Threat. Every Boost you add creates almost enough Advantage on average to offset the average Threat of a Difficulty die, but the Difficulty die creates Threat an additional 1 in 8 times more often than the Boost creates Threat.
Boost dice have the same odds to create double Advantage as Proficiency and Challenge dice have to create double Advantage/Threat, and will create double Advantage more often than Difficulty dice will create Threat, although it creates Advantage less reliably than a Difficulty die.
I’m rambling a bit, and not sure what my point is. I just enjoy dice math.