Dice and Probabilities

If I wanted to create a game system, one of the first things I’d tackle is the fundamental dice mechanic I’d like to use. Rolling dice, and the chances a player is supposed to have at succeeding different given tasks, obviously deeply influence many other design decisions – such fundamentals as attribute ranges and how they figure into the dice rolls, but all other aspects of the game system as well.

High or Low?

Do I want high or low rolls to be good? When you check a percentile number, you’ll want to roll 1d100 and get a result that is lower or equal to your target number – usually a skill, such as Accounting, Computer Use, or Melee Attack.

A system of rolling under a target value can be quite simple in other cases as well. If your attributes, such as Strength, Intelligence, and Charisma are basically on a 1-20 scale, you could roll 1d20 and aim for a result lower or equal to your attribute score. Since you generally will raise these values as your character progresses, rolls become easier over time.

However, I think that it is more intuitive to build a system where a higher roll is better. That is, you’ll want to generate a result that is higher than a target number. Under such a system, I would modify the dice roll with factors that affect the character – skill level, attribute bonus, conditional modifiers such as blindness, wound effects and so on – and the target number with modifiers that are external to the character: bad visibility, zero-G environment, strong rain, and so on.

Deviation

The two most straight-forward dice systems are 1d20, 1d20 – rolling a ten- or twenty-sided die, respectively – and 1d100 – percentiles, basically. Both are easy to grasp. If my target number is a, say, 10 or higher on 1d20, then I have a 55% chance of success. A +1 modifier increases this to 60%, +2 to 65%, and so on. Simple.

There is a lot of randomness in such a system, however. It’s equally likely for me to roll a 10 as it is to roll a 20 or any other of the 20 possible results. In other words, a d20 (or d100) has a high standard deviation; results are likely to be far from the median. A low standard deviation, conversely, means that results tend to be closer to the median value. In plain English this means that dice rolls are, in general, more predictable in this case, and character statistics tend to have a bigger impact on the result.

Examples of standard deviation:

Label 1d6 2d6 3d6 4d6 1d10 1d20 1d100 2d10 2d12 2d20 3d12
Variance 2,92 5,83 8,75 11,67 8,25 33,25 833,25 16,5 23,83 66,5 35,75
Std Deviation 1,71 2,42 2,96 3,42 2,87 5,77 28,87 4,06 4,88 8,15 5,98

In practical terms, a mechanic with a high standard deviation means that characters have a better chance to do really well – or really badly! A group of lowly Goblins could roll a number of 20’s and actually hurt the demigod-like player characters, who could roll a lot of 1’s. If you use, say, multiple d6, such high and low results are just much less likely.

If one wants a low standard deviation – and I do, since I prefer player actions and character stats over pure “dumb luck” – 1d6, 2d6, 3d6 and 1d10 seem good candidates. 1d6, 2d6 and 1d10 simply do not have a great range of possible outcomes and are probably bad choices.

But instead of picking blindly, let’s crunch some more numbers.

A player’s chances to roll a number X or higher are:

 X 2d6 3d6 1d20 2d10 2d12
1 100,00%
2 100,00% 95,00% 100,00% 100,00%
3 97,22% 100,00% 90,00% 99,00% 99,31%
4 91,67% 99,54% 85,00% 97,00% 97,92%
5 83,33% 98,15% 80,00% 94,00% 95,83%
6 72,22% 95,37% 75,00% 90,00% 93,06%
7 58,33% 90,74% 70,00% 85,00% 89,58%
8 41,67% 83,80% 65,00% 79,00% 85,42%
9 27,78% 74,07% 60,00% 72,00% 80,56%
10 16,67% 62,50% 55,00% 64,00% 75,00%
11 8,33% 50,00% 50,00% 55,00% 68,75%
12 2,78% 37,50% 45,00% 45,00% 61,81%
13 25,93% 40,00% 36,00% 54,17%
14 16,20% 35,00% 28,00% 45,83%
15 9,26% 30,00% 21,00% 38,19%
16 4,63% 25,00% 15,00% 31,25%
17 1,85% 20,00% 10,00% 25,00%
18 0,46% 15,00% 6,00% 19,44%
19 10,00% 3,00% 14,58%
20 5,00% 1,00% 10,42%
21 6,94%
22 4,17%
23 2,08%
24 0,69%

Let’s visualize this:

I think this beautifully illustrates the matter. You can see that a few median numbers cover a much larger probability range for multiple dice than for the linear 1d20 line.

So, a multiple-dice approach seems like the way to go. But which one?

Bonus Thoughts

This is where we begin to consider the consequences for the game system itself. 2d6 only has 11 possible outcomes, 2…12. This means that any +1 will shift success by quite a bit, but not every +1 has the same “value”. If your target number is to roll 7 or more on 2d6, that’s a 58.33% chance. A +1 to your die roll makes this 72.22% – Your +1 is “worth” just over 14 percent points. A second +1 would make your success chance 83.33%, so that second +1 is “worth” a little over 11%, and so on. Eventually, your sixth +1 bonus would only increase your chances by 3%, success would become automatic and further bonuses would be wasted.

This in itself is not bad, as you can easily argue that, realistically, in any given situation, once you have enough factors in your favor, success should all be assured. However it does mean that characters are described by a limited numerical range, and this limited variation means that they tend to become fairly similar. It also means that character progression is naturally limited. This can be a problem for long term games. Sure, there is always the possibility to reward players and their characters with other benefits – a castle, an alien artifact, favor from powerful allies, whatever – but I firmly believe that most players like to see the statistics of their character improve over time. It can be slow, but it must be noticeable and overall meaningful.

The number of possible results for some possible dice combinations are:

Label 1d6 1d10 2d6 3d6 2d10 1d20 4d6 2d12 3d12 2d20 1d100
# Results 6 10 11 16 19 20 21 23 34 39 100

The granularity, if you will, of 2d6 seems too limiting; 3d6, 2d10, 4d6 and, at the upper end, 2d12 seem reasonable.

Simplicity

In the end, you could scale a game system for any of these choices. However, there is also something to be said for simplicity. I will be picking 3d6 and rolling high as the standard dice mechanic. Not only do the 3d6 probabilities approximate a nice Gauss curve, six-sided dice are also by far the most common and using them removes one barrier of entry from the game design; even non-gamer players are likely to have them around and do not need to run to a hobby story to pick up weird polyhedral dice.

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