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Bell Curve or Linear?


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So, simple question is a 1d20 interchangeable with a custom 20 sided die with nineteen faces show 1, and one face showing 191?

If the answer is no, then rolls matter.

If the answer is some for of, "it depends," then rolls still matter.

If the answer is yes, then consequences don't matter.
I said the results matter. Dice rolls are plugged into game mechanics which then determine some result. And multiple different outputs of the randomizer can end up producing the same result when fed through the mechanics. And conversely, the same roll can produce different outcomes after being being processed - a lucky hit from a Vorpal sword might be a lot worse than one from wiffle ball bat. In DnD, people complain about save or die/suck spells, and not so much about save versus tiny penalty spells (unless its that those spells are bad compared to other ones). Something like the GM calling for a Perception test or Willpower roll just to spook the players are also situations where rolls don't matter because they all lead to a null result.

So if you need to roll X to hit, where X isn't 20, then obviously replacing everything but the 20 with a 1 makes a big difference. If I need a 13, then replacing 13 through 19 hoses my chance of doing something because it produces a different result. The change directly impacts how often I'll succeed. OTOH, if you actually do need a 20 (maybe the target is super tough; maybe you're playing the "Critical Hit: The dice pool game" where you roll a shit ton of d20 and only natural 20s count for successes and everything is null) then maybe the custom d20 with a 20 and 19 ones produces identical results to a normal roll.
0,0,0,0 = 1%
0,0,0,1 = 14%
0,0,1,1 = 70%
0,1,1,1 = 14%
1,1,1,1 = 1%

Now the ideal crops up 70% of the time. Within the confines of the odds of what happens in only four tests, we've significantly increased the odds that our RNG operates close to the ideal.
Seems more like a corner case because your mode accounts for the majority of the outcomes to me though. If you look at something like 3d6, it's most common outputs only clock in 12.5% each, with some 11.57s coming in right behind, while it's more common to roll a 11, it's still way more likely that a different number is rolled.

Umm, hmm. I think the quote above I pulled yesterday was then edited. But IIRC, you had an example where you were altering the distribution of your pass or fail tests. Maybe I'm just being dumb somehow, but I'm not sure how you actually change the distribution you get from multiple independent binary tests without changing the chance to succeed on any one test. Dice don't have memory by default. I'm not sure how you make that work.

Moreover, if the "ideal," as you put it, is that people get very close to the expected outcome over a series of rolls, then you should just use a deck of cards in your design instead of either a linear or bell curve dice system. If you have a deck with 2 successes and 2 failures, then you can make sure people ALWAYS get 2 successes and 2 failures every 4 draws (or do like 3/3 to cycle every 6 tests, or whatever).

Shadow Captain

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Not so much. A true 2d4 distribution gets an inverse V distribution, as does 2dN for any standard dice. 2d{2, 3, 3, 4, 4, 5} produces a somewhat different distribution though, because the probability of 2 and 5 are lower on the individual die. Because the input die aren't a uniform distribution but are center weighted that causes more of a curve to build up, much the way that adding a center weighted distribution like 2d4 to another 1d4 does.
While the final curve might not have rounded edges, the deviations are pretty close. Instead of 3 standard deviations giving you a spread of 68/95/99.7% of all possibilities, you do get a 72/94/100% spread. For purposes of rpgs, it means you have a no and, no, no but, undecided, yes but, yes, and yes and resolutions. More importantly, it means that you have greater predictability while maintaining the random factor. In other words, it allows for the high end characters to reasonably measure their ability to storm the castle while still allowing for the Bard shot against Smaug.
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