A normal distribution (a.k.a., a “bell curve”) is a reasonable match for how many things that have a partially random outcome turn out. Most often, values will be close to the mean, more rarely to one extreme or the other and very rarely extraordinarily so. In rolling the outcome of a melee attack for example, the mean would be about how well on average one would do and the variance the best and worst one could possibly do. However the best way to simulate a normal distribution is to sum a large number of random outcomes, and without computerized help this isn’t convenient to do. So rather than having dice rolls simply give you a linear outcome, what’s a good way to do normal outcomes in role-playing?
We’re talking table-top RPGs here? Although rolling one die at a time will give you a uniform distribution, two or more dice rolled at the same time gives a normal distribution.
For example:
Rolling two dice gives you a sample space 2-12 normally distributed around 7
Rolling three dices gives you a sample space 3-18 normally distributed around 10.5
You’re right, even a few die give a (roughly) normal distribution. I was thinking of the widespread practice of rolling a single D20 to determine outcomes. Do they just use the numbers 1-20 normally or are the numbers divided into groups representing percentile deviations?*
*if I’m saying that right, I have only a passing acquaintance with statistics
Most role-playing games use a single linear d20, or something even simpler, but there are optional rules for D&D (which could be adapted to other RPGs) that replace all of the d20 rolls with 3d6. This leads to some odd cases in corner situations where the rules really weren’t designed for it, but only a very few, and overall actually works pretty smoothly (possibly even more smoothly than the default rules, in some contexts).
A single D20, e.g., an icosahedron, will give a uniform distribution across 1, 2, 3, …, 20 and not a normal distribution. That is, on a D20, a “1” has the same chance as appearing as a “20”, and both of those have the same chance of appearing as a “10”.
Let’s consider what Chronos suggested with the optional D&D rules. Forgive me if this confuses the issue, but let’s compare rolling a D12 (dodecahedron) versus the sum of two D6’s (summing two 6-sided dice). Let’s skim over the part that you can’t get a “1” on two D6’s (let’s say you re-roll if you get a “1” on the D12). Both of the dice rolling systems give the range between 2-12. On the D12, the “2” and the “7” and the “12” all have equal probability of occurring. But on the 2D6, a “7” is more likely than snake eyes or a “12”. The 2D6 has more of a normal-ish distribution than the “uniform” distribution of the D12 that’s missing the “1” face.
So Lumpy, you’re correct about summing dice as a way to impose a normal distribution. If you’re trying to simulate something with a range of, say, 10-130, you could consider summing a whole pile of dice, or you could do a thing like rolling 2D6 and then take the result and use algebra to transform it to cover the range:
Result of 2D6 must fall between 10 and 130, so:
10 <= 13 * 2D6 - 16 <= 130,
and it’s somewhat normally distributed (or at least more normal than a uniform distribution).
One of neat ways I have read to get a bell curve from using D20 is to roll three of them, and take the median roll. (So if you roll a 3, 12 and 7, your final roll is 7). This way you can still cover the fringe cases (getting a 1 or 2, or a 19/20) easily, but boy aren’t those people who rely on Critical Hits going to be pissed.
Well, there’s lots of systems that avoid all the hassle by using 3d6 or something similiar for the skill rolls, like GURPS. Or a big pool of dice, but I’ve never tried those. GURPS seems to work alright for our WH40k Inquisition game, though it did take a long time for our GM to tune psionic powers after we converted from Dark Heresy system.
If our veteran adepts are trying something reasonably easy like firing an aimed burst at a target 10m away with a Hellfire rifle on autofire, the chances of totally missing are less than 2% (17 would be a miss and 18 a fumble) and that’s how it should be. In addition, because the number you succeed by counts (rolling 2 under the target number gives a total of 3 hits with the Hellgun for example), it’s far less random than D&D combat. Skilled people both hit more often and harder with great consistency, and while mooks can sometimes get a lucky crit in they tend to only land one or two shots if they hit at all normally. I didn’t really like the system at first but after being forced to play it for a few years now I’m starting to slowly appriciate it.
Champions/Hero System also used 3d6 for to-hit and multiple d6 for damage to generate a roughly normal distribution. One benefit of such a system in a superhero game is that once a minion’s chance to hit or damage dealt fell sufficiently behind a given superhero’s power level, the threat posed by the minion fell off dramatically, making it possible but extremely unlikely for him to threaten the superhero. This worked fairly well to simulate the sort of mismatches that occur in superhero comics.
Conversely, I played a board wargame (Ironclads) about heavily armored Civil War ships with cannons, that used a normal distribution to simulate penetrating armor and to judge “critical hits”, but for reasons known only to the game’s creators, had another category of slightly less destructive “special” hits that used a uniform distribution (any “6” on a d6).
The end result of this system was that big guns at close ranges were needed to penetrate heavy armor and destroy a ship…but any guns at any range could disable it with “special” hits. Victory usually went to whichever side was better able to drench enemy vessels in a rain of lightweight projectiles and wait for special hits to occur. This, in case you are wondering, was generally ahistorical and produced some pretty bizarre results.
You can also approximate the effects of a normal distribution with a single die by baking the distribution into the table you’re rolling on.
Say you’ve got an ability vs. difficulty roll. You have the player look up the required d20 value in a table that cross-indexes ability vs. difficulty.
The “level 5” skill column would look something like this:
Difficulty --> d20 Roll
1 --> 1
2 --> 4
3 --> 7
4 --> 9
5 --> 10
6 --> 10
7 --> 11
8 --> 11
9 --> 12
10 --> 14
11 --> 17
12 --> 20
While the “level 9” skill column would look something like this:
Difficulty --> d20 Roll
1 --> 1
2 --> 1
3 --> 1
4 --> 1
5 --> 1
6 --> 4
7 --> 7
8 --> 9
9 --> 10
10 --> 10
11 --> 11
12 --> 11
FUDGE and FATE use a system where you are given a rating for a skill (Terrible, Poor, Mediocre, Fair, Good, Great, and Superb) and then you roll 4d3, where each side of the d3 is a +, -, or blank. (Usually it’s 6-siders with two of each symbol on opposite sides.) You add up the result and shift your current rating up or down based on the result. (E.g. +,+, -, blank on the four dice would give you a net +, so your rating shifts up one). It’s numerically equivalent of rolling 4d3 and subtracting 8.
And then there are the dice pool systems such as Wild Talents, a d10 based system. Instead of going for normalization, they give the user choices of how they want a skill to work. You roll the number of dice in your pool, and you must have at least one pair of the same number to succeed. If you have multiple matching numbers, you choose the number you succeed with. The higher the number, the better the success. The more matches you have for a given number, the faster the success.
So for example, if you are pressed for time and you are trying to pick an easy lock, in your pool of matches you have 4 3’s, 2, 6’s and 2 0’s (treated as 10). You’d probably choose 4 3’s over a pair of 0’s, because you probably only need a level-3 success to unlock it, and the 4 matches will speed you up considerably. However, if you have to dismantle a pressure sensitive switch or something similarly difficult, and have plenty of time, you want the better success those 0’s will bring you.
And then there’s a success with 1’s. 1’s are quirky successes. You succeed, but the GM generally will have something odd result, which can be good or bad. Just don’t roll all 1’s, since that will cause a critical failure.
If you want to simulate a random distribution, you could print out tables for any situation that could occur, then roll a D10000 and look up the results in the table. You roll a D10000 by rolling 4 10-sided dice and read them in a specific order (If you’re from America a good rule of thumb is Red-White-Blue-Green, the first three of which has an easily-remembered mnemonic.)
Anything that happens less often than 1 in 10000 cases isn’t worth worrying about, or you could print out extended tables and roll on them when you roll a 1 or 10000 
Of course, printing out dozens and dozens of tables for how many D20 rolls out of 15 will hit when you need a 5, then another table for how many out of 10 will hit needing 19, etc etc could be very time consuming as well as looking them up.
Incidentally, the reason why normal distributions show up so often in the first place is precisely the fact that a normal distribution is approximated by a bunch of dice added together. Or rather, if the outcome of something is determined by the sum of a large number of random variations of all approximately the same size, the result will follow a normal distribution.
The Hamster King, that method works, but it requires a heck of a lot of tables and look-ups, which takes up both time at the table and pages of the rulebooks.
One game that used percentile divisions with uniform dice rolls was James Bond 007. Pretty much everything was two D10 rolls. It had a very neat way to add “movie magic” : Certain actions could be awarded “Hero Points” - these were meant to be pretty rare, with only a few earned per mission (however, the better you did, the more you were likely to get). A Hero Point could be expended to either shift a result into a better bracket, or force a re-roll. Thus, instead of getting crushed by the falling bus, your agent might slip away at the last possible second. Or if they needed that one perfect shot to knock the detonator out of the villain’s hand but the roll says they blew it, they’d get another chance at it.
Some versions of the D&D rules (for instance, those used in the Eberron campaign setting) have something similar to that, called “action points”. I’m not a big fan of them, but that’s more of a personal style thing than a fundamental objection.
It depends. Looking up a roll in a table takes time, but so does summing dice, particularly if the player is going to be making the same roll repeatedly.
It also gives the designer finer control over the probabilities (particularly if its driven by % dice instead of a d20). This makes game balance easier – you can nudge values slightly higher or lower without having to completely renormalize to a new rolling mechanic.
I agree that you wouldn’t want to use a baked-in table for everything. But then, you probably shouldn’t even be using normal distributions for *most *things. You don’t need the low-probability tails on the high and low ends for a roll that you only make occasionally.
I like this. The probability distribution is roughly a parabola with a peak at 10.5 and zeros at 0.5 and 20.5. The probability of a 10 or 11 is about 7.5%, and the probability of a 1 or 20 is about 0.7%.
You might object that a parabola doesn’t have the tails like a true normal distribution has, but if you’re working with a game designed for a more uniform distribution, you probably don’t want to make correct but highly unlikely tails anyway.
Generic “you”, not Crowbar of Irony +3.