Mathematicians help with dice probabilities!

[Morrus]

I have a small design problem. In line with the toolkit nature of my game, I want to include a small sidebar with a couple of alternative dice resolution mechanics. The probabilities need to remain the same; just the dice and how you read them need alternative versions.

The current system: roll a d6 dice pool, add totals together, roll over a target figure. The target figures look like this:

[TABLE="width: 500"] [TR] [TD]Trivia[/TD] [TD]-[/TD] [/TR] [TR] [TD]Easy[/TD] [TD]7[/TD] [/TR] [TR] [TD]Routine[/TD] [TD]10[/TD] [/TR] [TR] [TD]Challenging[/TD] [TD]13[/TD] [/TR] [TR] [TD]Difficult[/TD] [TD]16[/TD] [/TR] [TR] [TD]Demanding[/TD] [TD]21[/TD] [/TR] [TR] [TD]Strenuous[/TD] [TD]25[/TD] [/TR] [TR] [TD]Severe[/TD] [TD]29[/TD] [/TR] [TR] [TD]Herculean[/TD] [TD]33[/TD] [/TR] [TR] [TD]Superhuman[/TD] [TD]37[/TD] [/TR] [TR] [TD]Impossible[/TD] [TD]40[/TD] [/TR] [TR] [TD]Mythical[/TD] [TD]45[/TD] [/TR] [/TABLE] Here are some alternatives I'm considering including in that sidebar:

[h=4]Count Dice Rolling 4+[/h]
A number of systems do this. This makes each dice 50% chance of rolling 0 or 1, so the average dice rolls 0.5.

The average roll on the default method is 3.5. Does this mean all I need to do is divide target numbers by 7 to retain the correct probabilities? You'd have to round up, I guess, meaning difficulty numbers range from 0-7.

[h=4]Count Dice Rolling 5+[/h]
Assuming the above method is correct, this results in dividing target numbers by 10 instead of 7. Difficulty numbers range from 0-5, so it lacks granularity.

[h=4]Count Dice Rolling 6+[/h]
I don't think this one is viable. It's useful because you can buy d6s with a different "six" face, so it's visually easy. But it reduces granularity massively, dividing difficulties by 21.8 (call it 20). This makes for only a handful of difficulty numbers (0-3, basically). I think this can be discounted.

[h=4]1d20 + bonus[/h]
A bit of a departure down D&D Highway. I'm not sure how to work out the equivalent here.

If we assume 1d20 rolls 10 on average (rounding down for simplicity). So that's our baseline for no bonus, or our "routine" entry in the table above.

Each d6 needs to convert to a bonus. Each rolls 3.5 on average. Let's call it 3 (rounding down again). So the roll is 1d20 + (3 for each die). Using this rolling method, the difficulty targets remain the same.

Is that correct?

[h=4]Percentile Dice[/h]
I find percentile dice can add a more technical/scientific feel to a game. So, how to convert here?

Average roll is 50. So that's our Routine entry.

Ugh. I'm stuck. My brain won't go where it needs to go! So 100 is five times 20, and on d20+mod each d6 converts to +3. So on d100, each d6 converts to 15? And the difficulties are five times higher? So 5d6 vs. Difficult [16] converts to d100+75 vs. 80?

Nah, that's not right at all!

 

[steenan]

You must remember that the probability distribution is two-dimensional here - probability of success depends on the difficulty level and on skill level. It is very hard to keep it reasonably intact when translating to a different resolution mechanics.
In other words, you will have to decide what exactly you want to keep and what can change with the change of dice.


A pool of d6 with success on 4+ definitely has less granularity than a pool of summed d6, but for a reasonable number of dice rolled (let's say, 4 and up) it is possible to keep the difficulties and skill levels equivalent.


With success on 5+ or on 6 you not only change the expected number of successes, but also the shape of the distribution. The bell curve you get is no longer symmetrical; the mean is moved to lower values, but you have a long "tail" towards high number of successes.
In other words, the rare lucky rolls are much better compared to the average than they are in a symmetric distribution. Maybe that's something you want, maybe not.
If you want to use this kind of roll in your system, you should probably re-scale the skills.


The d20+bonus makes it much harder to get any kind of equivalency. You may easily set the difficulty levels to get the same probabilities for a fixed value of skill bonus, but you won't be able to get correct probabilities for different difficulties and skill levels. If all characters are at a similar skill value, that may not be a problem. But if you want a wide range of skills to matter (and that's what I suspect, judging from the wide range of difficulties), it's going to be hard.
Probably the best you can aim for is keeping the difficulty-skill equivalency as it was (with the target numbers you gave, 2 dice have around 50% at easy, 3 at routine etc.).


With percentile roll-under you have to think how you'd like to represent the skill values and difficulty levels. Unless you use some fancy math to calculate the success threshold from skill and difficulty, you won't get anything resembling the dice pool results.


It is easier to switch resolution systems if you aim for a specific behavior. If you want general equivalency, it's very hard to do.

 

[Morrus]

You've certainly identified the difficulties I'm having well!

Without doubt different mechanics make for very different play experiences (which is why folks like particular mechanics). They alter the curve, granularity, speed of play, and more.

The best I'm hoping for is a sidebar which introduces each and explains how each will be different. The actual difficulty levels will, of course, need reconfiguring for each.

Here's what I came up with so far.

http://www.woinrpg.com/alternative-dice-rolling-methods/

It's a reasonable start, I think. The percentile stuff is way off though. I couldn't wrap my head around it.