How can I compare the dice ranges for my game features to understand the minimums and maximums?

Question

I have a lot of weapons on my own system and I want to split on four categories like, common, uncommon and stuff.

I want to split them by how good these weapons perform damage wise. But I have difficulties with the underlying math, since some weapons have very close maximum and minimum damage numbers.

For Example:

• 2d12+3: can roll values between 5 and 27
• 3d8+2 can roll values between 5 and 26

I have many different weapons with different dices in different quantities. The dice range from d4 to d12 and the count from 1 to 3.

Ex: 1d4+3 / 3d6+2 etc.

I am looking for how to compare my weapons with each other damage wise, so not only for the examples but also how to do the comparsion in general.

Answer 1

I mean, if you want the best answer,

Take a basic probability and statistics class.

That’s how you figure these things out, with basic probability and statistics. These are taught in most high schools and pretty much all colleges (barring extremely specialized schools, maybe), plus there are a plethora of online courses out there.

For the very most basic things, a single (fair) die has what’s called a “uniform distribution.” That is, when you roll a d6, you have an even (“uniform”) chance of getting 1, 2, 3, 4, 5, or 6. In probability and statistics, we talk about the “expected value” of distributions, which is what you can expect, on average, from that distribution. For a uniform distribution, the expected value is just the average (arithmetic mean) of each possibility in it. For a d6, that’s

$$ \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5 $$

In a probability and statistics class, you’d learn about the expected value for all sorts of things. You’d also learn shortcuts for calculting them. In the case of a regular dX die (numbered 1 through X), the expected value is (X+1)/2, or (equivalently) X/2+½. And so for the standard dice:

Die	Expected Value
d2	1½
d3	2
d4	2½
d6	3½
d8	4½
d10	5½
d12	6½
d20	10½

For multiple dice that are added together (e.g. 4d6), you can determine the expected value by just multiplying the expected value by the number of dice (e.g. 4×3½ = 14). This still works when you use dice of different sizes (e.g. 4d6 + 3d4 = 4×3½ + 3×2½ = 21½). Flat numbers have an expected value of themselves, so you can just add those in (so 4d6+3 has an expected value of 14 + 3 = 17).

And that basically covers simple rolls of dice where you just add everything together. Note that there is a lot more to be said here. For instance, while you can just add expected values together to get the expected value from adding dice together, the sum of several dice does not produce a uniform distribution and 3d6 produces very different results from 1d20 even though they are both expected to average out to 10½ over the long run.

As for minimums and maximums, that’s going to depend a lot on how exactly you’re defining those. But if, say, you apply those to individual dice, so for example “d6, minimum 2,” and you’re just rerolling numbers below the minimum or above the maximum, you’re looking at a set of 2, 3, 4, 5, 6. So you can just do (2 + 3 + 4 + 5 + 6)/5, which equals 4. As long as your minimums and maximums maintain that uniform distribution (each number appears no more than once), all of the above will hold true, you just need to figure out the expected values of individual dice.

On the other hand, if you apply the minimums and maximums to the sums of multiple dice, so something like “3d6, minimum 8,” then you’re not dealing with a simple uniform distribution anymore, and that gets more complicated to calculate. Not necessarily very complicated, but you need more than the above. Likewise, if instead of rerolling numbers below minimums or above maximums, you bump numbers up or down as needed, that will also change things (since, for example, “d6 minimum 2” would be twice as likely to have a result of 2 as any other number). For specific, more complex rolls, separate questions specifically about that roll can be a good way to learn, but to be able to do it yourself, again, I recommend finding a decent online course.

Answer 2

While I agree with KRyan's answer, I'm going to suggest something simpler:

Look at probability distributions.

AnyDice is a good (and free!) tool for this. Put output 3d6 into the text box and hit "Calculate", and you'll see something like this:

The length of each bar indicates the probability of rolling that result. You can see that the minimum here is 3 and the maximum is 18.

Now replace that with output 1d16+2 and hit "Calculate" again.

Once again you have a minimum of 3 and a maximum of 18, but the distribution is very different! This is formally called a uniform distribution, with every result being equally probable, while the 3d6 is closer to a normal distribution, with the middle values being much more likely than the extremes.

If you don't have much experience with probability or statistics, I think looking at the distributions in nice bar graphs like this is a good way to get an intuition for them. AnyDice can also tell you statistical measures like the mean and standard deviation, but if you don't have experience working with those, the bars show you the same thing in an easier-to-eyeball way.

P.S. AnyDice in particular can also handle all sorts of complicated dice systems beyond standard D&D XdY+Z, which is one reason I recommend it in particular.

Answer 3

As the other answers already explored the mathematical and a tool supported approach I want to give a less "mathy" rule-of-thumb approach.

Using dice pools with sameish maximum and minimum values both are on avarage equally strong. However, you can compare a weapon with less high value dice (e.g. 1d20, 2d12) with a weapon with more low value dice (e.g. 4d6, 8d3) this way:

The weapon with more dice will more likely roll results close to the avarage, while the weapon with less dice will (in comparsion) have a higher probability to roll the extrem values close to maximum and minimum. Note however that all dice-pools with more than 1 die will always have a higher probability to roll the avarage value, than to roll min or max.

So to look at the weapons you gave for an example:

• 3d8+2
  • more reliably rolls close to avarage results (53% chance to roll 13 to 18)
  • less likely to roll high damage results (4% chance to roll 23 to 26)
  • less likely to roll low damage results (4% chance to roll 5 to 8)
• 2d12+3:
  • less reliably in the "avarage zone" (44% chance to roll 13 to 18)
  • more likely to roll high damage results (10% chance to roll 23 to 27)
  • more likely to roll low damage results (7% chance to roll 5 to 8)

This effect gets much more extreme when comparing dice-pools that differ even more in the dice-count (for example 1d20+4 vs 6d4).

Which of the two is better is highly situational and also depends on the system you are playing.

For example the 3d8+2 weapon would be better than the 2d12+3 to finish off an opponent with 11 hp left (89% vs 85%), while the 2d12+3 weapon has better chances in one-hitting an opponent with 23 hp left (4% vs 10%).