Dice Pool Roll Under

Rolling under your abilities.

One of the most controversial of ruling suggestions in B/X - nested snugly into the subsection, Dungeon Mastering as a Fine Art on page B60 under the moniker that There's always a chance. This rule has become a default mechanism for some referees - it's simplicity and quickness at the table while differentiating characters more or less naturally capable luring them in - and has become anathema for others: it's emphasis on abilities defying the class-level focus of almost any other challenge in the game, it's swingy potential for a well-endowed character to fail while an incapable character succeeds (the "I loosened it up for you" hypothesis) puts them off. However - in recent vintage - I've come across a variant which may help to alleviate some of the tensions - rolling under using a d6 dice pool.

My interpretation as follows:

Pooling Under your Abilities

When presented with a task - skill, talent, or otherwise - not covered by a rule but requiring a roll to account for potential of failure, the referee should, having considered the described manner of resolution, ask the player to roll a number of d6 proportional to the difficulty of the task compared to the efficacy of the proposed resolution mechanism:

• Easy tasks, or tasks well achieved, roll 1d6.
• Normal tasks, or perhaps easy tasks the method of resolution provided being questionable, roll 2d6.
• Challenging tasks, or tasks being approached poorly, roll 3d6.
• Difficult tasks, or tasks being approached entirely wrong, roll 5d6.
• Impossible tasks, or descriptive equivalent, roll 8d6,
  

If the result of the pool exceeds the ability score of the character rolling, the task fails. Otherwise, the task succeeds.

Optionally, to represent the off-chance of failure even when doing something simple (or, I Missed My Mouth With The Cup syndrome), use single-explosion dice. Thus, an Easy task has a likely result of 1-5, but an unlikely result of 7-12; or a Difficult task has an average result of 17 or 18, but a potential (and highly unlikely) result of as much as 60.

Credit where Credit is Due

Note, as is most of the time with house rules, I did not originate this idea. Googling it to find the page on which page the "roll under" rule in B/X was, I found several social media platforms where this or a similar system was being discussed. Credit where credit is due, I am writing this article listening to the Biggus Geekus podcast. Kudos - Randy and Joe, for the interesting topic - and credit, DM Dastardly Dad for the great suggestion!

Why write about it then, if it's already so popular, as you say? I think it's a neat idea - but what I didn't see was a detailed breakdown and mathematic scrutiny. Also - it's a great idea: why wouldn't I amplify it?

How Do the Chances Look?

Without the optional exploding die rule, the average roll for each of the respective pools are as follows:

	Minimum	Average	Maximum
1d6 )	1	3	6
2d6 )	2	7	12
3d6 )	3	10	18
5d6 )	5	17	30
8d6 )	8	28	48

Thus, an average character (ability score 10) cannot fail an Easy task, would only fail ~8.3% of the time for a Normal task (requiring an 11 or 12 on 2d6), and would conversely fail ~47.2% of the time for a Challenging task. With this system, it becomes difficult to be impossible to pass: however the chance of failure changes rapidly as the curve becomes wider and wider. Also of interest - a character with a maximum score will never fail for Challenging tasks or below; where a character with a minimum score has at least a 50% chance for an Easy, the minimum score on the standard 3d6 stat array being 3.

To expand, a character with a high or low ability has a chance to succeed on this scale as follows:

Ability	Easy	Normal	Challenging	Difficult	Impossible
V. Low (4)	66.7%	16.7%	1.9%	0%	0%
Low (7)	100%	58.3%	16.2%	0.26%	0%
Avg (10)	100%	91.7%	50%	3.23%	0.03%
High (13)	100%	100%	83.8%	15.2%	0.11%
V. High (16)	100%	100%	98.1%	40%	0.78%

The numbers chosen were chosen for their relationship to the assumed 3d6 curve of character abilities: 13 is one standard deviation above, 16 two standard deviations above, 7 one below, and 4 two below. You could create a spreadsheet function to produce more data - but the above, I think, provides a good illustration.

Additionally, the percentiles are estimated - rounding conveniently for my brain - especially to the "Impossible" side of the scale: as rolling numbers less than 11 is almost statistically implausible: the likelihood of hitting those numbers being percentages of a percentage point. A character with an 18, for example - the nominal maximum in an OSR game - only has a 2.37% chance to succeed at an Impossible task.

Introducing the exploding mechanic makes the math substantially more difficult. So - thank you, AnyDice, for doing it for me! The likelihood of success, based on high or low ability score, with exploding d6 (depth 1: that is, if you roll a 6 on your explosion roll, that second 6 does not, itself, explode):

Ability	Easy	Normal	Challenging	Difficult	Impossible
V. Low (4)	66.67%	16.67%	1.85%	0%	0%
Low (7)	86.11%	58.33%	16.2%	0.27%	0%
Avg (10)	94.44%	91.67%	50%	3.24%	0.01%
High (13)	100%	97.22%	83.8%	15.2%	0.08%
V. High (16)	100%	97.69%	98.15%	39.97%	0.74%

At first glance, the numbers seem somewhat similar in some places. And this makes sense - for example, a character with an Ability score of 4 has no impact to an Easy task: if you roll a 6, eligible to explode, it doesn't matter: the roll has already failed and no explosion is necessary - thus, the chance of success is unmodified. However, it does introduce increased failure (albeit unlikely) for any other of the ability bands calculated: a score of 7, for example, on an Easy task would, if an initial 6 is rolled, only have a 1-in-6 chance to succeed as any other roll on the explosion would push the result over the ability score and into failure territory.

Regarding the range and average of task rolls, incorporating the explosion mechanic, consider as follows (again, courtesy AnyDice):

	Minimum	Average	Maximum
1d6 )	1	4.08	12
2d6 )	2	7.19	24
3d6 )	3	10.55	36
5d6 )	5	17.5	60
8d6 )	8	28	96

At a glance, the higher dice pools seem less phased by explosions than the lower ones, in terms of averages. That does make some sense - as the likelihood of the higher number of explosions, and thus higher results, becomes vanishingly small as it increases.

Curiously - looking at some of the intermediary probabilities of success chances, it becomes impossible to have certain results. On an Easy check - 1d6 - obviously, you can never have a result of 6: because there will always be at least a +1 from the second die rolled. AnyDice, however, claimed that a 12 was also impossible on 2d6, or a Normal task, with exploding dice. At first, it seems plausible - there is only one way to get 12 on 2d6: two 6s, which would explode - but then if you think about it: you could have one 6, exploding by 2 to 8, and then roll a 4 on the other die: 8+4 is 12: so, at least one way to get a 12 on 2d6. I wonder if that's a glitch or if I simply entered an improper command. Similarly, I wonder if this would have impacted calculating the means...

On a final note, some of the probabilities at higher challenge tiers appear to be very close: which I would chalk up more to differences in how AnyDice rounded the results than to actual probability differences. Regardless - the chart does provide a good view into the way success or failure appears when using this method.

Conclusion

I like this. I could see myself using this rule with the explosion variant. It's less swingy than 1d20, but still makes abilities matter. Likewise, the explosion acts as an equalizer: producing a setup where you can fail - but the likelihood of that failure is largely dependent on your ability and doing the task right. 

There is room for some interpretation in the "would X work?" - speaking to defining the mechanism of the task - but that's what refereeing is all about. I ... probably would not call for any Impossible checks though. Waste of dice time.

Delve on, readers!

Public domain artwork retrieved from OldBookIllustrations.com and adapted for thematic use. Attribution in alt text.