D&D: Odds for Xd6 Roll Under Ability Score Checks
Ability score checks seem to be of those things people either love or hate. I personally avoid them as much as possible. As long as player characters have enough time and appropriate resources, they can make it. And even when there is something to roll, I first look to saves or some other existing resolution systems. Then, and only then I'd consider rolling an ability score check using the Judges Guild system (d00 roll under, doubles result in injury).
Whichever game I play or run, I like to know (or have a feeling for) the odds. This enables me to make educated decisions about the risk, which in turn helps me decide what to do (or how to adjust difficulty if I'm the Judge). With d20 and d00 it is quite easy: a pip is either 5% or 1%. +/-1 then is either +/-5% or +/-1%. But the moment we introduce dice pools, the uniform distribution is gone and we get normal distribution instead. What are the odds then?
With that being said, Xd6 systems have been present since the beginning (e.g. loyalty and morale check). Robert Fisher house rule was inspired by The Fantasy Trip's system which basically adds or removes dice based on how challenging something is.
Here are the odds of rolling ability score or less on Xd6 (rounded to two decimals):
| ABILITY SCORE | 1d6 | 2d6 | 3d6 | 4d6 | 5d6 | 6d6 | 7d6 | 8d6 |
|---|---|---|---|---|---|---|---|---|
| 3 | 50.00% | 8.32% | 0.46% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |
| 4 | 66.68% | 16.65% | 1.85% | 0.08% | 0.00% | 0.00% | 0.00% | 0.00% |
| 5 | 83.35% | 27.76% | 4.63% | 0.39% | 0.01% | 0.00% | 0.00% | 0.00% |
| 6 | 100.00% | 41.64% | 9.26% | 1.16% | 0.08% | 0.00% | 0.00% | 0.00% |
| 7 | 100.00% | 58.30% | 16.20% | 2.70% | 0.27% | 0.02% | 0.00% | 0.00% |
| 8 | 100.00% | 72.18% | 25.93% | 5.40% | 0.72% | 0.06% | 0.00% | 0.00% |
| 9 | 100.00% | 83.29% | 37.50% | 9.72% | 1.62% | 0.18% | 0.01% | 0.00% |
| 10 | 100.00% | 91.62% | 50.00% | 15.90% | 3.24% | 0.45% | 0.04% | 0.00% |
| 11 | 100.00% | 97.17% | 62.50% | 23.92% | 5.88% | 0.99% | 0.12% | 0.01% |
| 12 | 100.00% | 99.94% | 74.07% | 33.56% | 9.80% | 1.97% | 0.28% | 0.03% |
| 13 | 100.00% | 100.00% | 83.80% | 44.37% | 15.20% | 3.59% | 0.61% | 0.08% |
| 14 | 100.00% | 100.00% | 90.74% | 55.63% | 22.15% | 6.08% | 1.21% | 0.18% |
| 15 | 100.00% | 100.00% | 95.37% | 66.44% | 30.52% | 9.65% | 2.21% | 0.38% |
| 16 | 100.00% | 100.00% | 98.15% | 76.08% | 39.97% | 14.46% | 3.79% | 0.74% |
| 17 | 100.00% | 100.00% | 99.54% | 84.10% | 50.00% | 20.58% | 6.12% | 1.37% |
| 18 | 100.00% | 100.00% | 100.00% | 90.28% | 60.03% | 27.94% | 9.39% | 2.37% |
Now, let's narrow it to four categories suggested by Robert (rounded):
| ABILITY SCORE | EASY 2d6 |
MODERATE 3d6 |
HARD 4d6 |
VERY HARD 5d6 |
|---|---|---|---|---|
| 3 | 8% | 0% | 0% | 0% |
| 4 | 17% | 2% | 0% | 0% |
| 5 | 28% | 5% | 0% | 0% |
| 6 | 42% | 9% | 1% | 0% |
| 7 | 58% | 16% | 3% | 0% |
| 8 | 72% | 26% | 5% | 1% |
| 9 | 83% | 38% | 10% | 2% |
| 10 | 92% | 50% | 16% | 3% |
| 11 | 97% | 63% | 24% | 6% |
| 12 | 100% | 74% | 34% | 10% |
| 13 | 100% | 84% | 44% | 15% |
| 14 | 100% | 91% | 56% | 22% |
| 15 | 100% | 95% | 66% | 31% |
| 16 | 100% | 98% | 76% | 40% |
| 17 | 100% | 100% | 84% | 50% |
| 18 | 100% | 100% | 90% | 60% |
If I were to use the above, I'd probably include the table in the house rules document so players can make an informed decision. And for me as well, so I can gauge difficulty. Although there is some allure for the notion of “simply” adding d6 dice for how challenging something is, I think I'll stick to my approach of first hearing out how players want to do something, and then picking the most appropriate resolution mechanism.
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